Finite Group DatabasebetaFinite Group Databasebeta| order | 32768 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C214 · Schur multiplier (H2) = C290 |
| extraspecial | 2+1+14 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C214 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
4073782660872416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 14 generators on 256 points:
(2,10)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(125,180)(126,181)(127,182)(128,183)(184,220)(185,221)(186,222)(187,223)(188,224)(189,225)(190,226)(191,227)(192,228)(193,229)(194,230)(195,231)(196,232)(197,233)(198,234)(235,248)(236,249)(237,250)(238,251)(239,252)(240,253)(254,256);(1,2)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,10)(17,38)(18,44)(19,45)(20,46)(21,47)(22,48)(23,39)(24,49)(25,50)(26,51)(27,52)(28,40)(29,53)(30,54)(31,55)(32,41)(33,56)(34,57)(35,42)(36,58)(37,43)(59,94)(60,95)(61,96)(62,97)(63,98)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,99)(75,100)(76,101)(77,102)(78,119)(79,120)(80,121)(81,122)(82,123)(83,124)(84,103)(85,104)(86,105)(87,125)(88,126)(89,127)(90,106)(91,107)(92,128)(93,108)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)(135,170)(136,171)(137,172)(138,173)(139,184)(140,185)(141,186)(142,187)(143,188)(144,189)(145,190)(146,191)(147,192)(148,193)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,194)(156,195)(157,196)(158,197)(159,180)(160,181)(161,182)(162,198)(163,183)(199,220)(200,221)(201,222)(202,223)(203,224)(204,225)(205,226)(206,227)(207,228)(208,229)(209,235)(210,236)(211,237)(212,238)(213,239)(214,230)(215,231)(216,232)(217,233)(218,240)(219,234)(241,248)(242,249)(243,250)(244,251)(245,252)(246,254)(247,253)(255,256);(3,17)(11,38)(18,59)(19,60)(20,61)(21,62)(22,63)(44,94)(45,95)(46,96)(47,97)(48,98)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(184,220)(185,221)(186,222)(187,223)(188,224)(189,225)(190,226)(191,227)(192,228)(193,229)(209,241)(210,242)(211,243)(212,244)(213,245)(235,248)(236,249)(237,250)(238,251)(239,252)(246,255)(254,256);(1,3)(2,11)(4,18)(5,19)(6,20)(7,21)(8,22)(9,17)(10,38)(12,44)(13,45)(14,46)(15,47)(16,48)(23,59)(24,64)(25,65)(26,66)(27,67)(28,60)(29,68)(30,69)(31,70)(32,61)(33,71)(34,72)(35,62)(36,73)(37,63)(39,94)(40,95)(41,96)(42,97)(43,98)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(74,129)(75,130)(76,131)(77,132)(78,139)(79,140)(80,141)(81,142)(82,143)(83,144)(84,133)(85,134)(86,135)(87,145)(88,146)(89,147)(90,136)(91,137)(92,148)(93,138)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,209)(156,210)(157,211)(158,212)(159,205)(160,206)(161,207)(162,213)(163,208)(174,220)(175,221)(176,222)(177,223)(178,224)(179,225)(180,226)(181,227)(182,228)(183,229)(194,235)(195,236)(196,237)(197,238)(198,239)(214,241)(215,242)(216,243)(217,244)(218,246)(219,245)(230,248)(231,249)(232,250)(233,251)(234,252)(240,254)(247,255)(253,256);(4,23)(12,39)(18,59)(24,74)(25,75)(26,76)(27,77)(44,94)(49,99)(50,100)(51,101)(52,102)(64,129)(65,130)(66,131)(67,132)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(109,164)(110,165)(111,166)(112,167)(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(155,214)(156,215)(157,216)(158,217)(184,220)(185,221)(186,222)(187,223)(188,224)(189,225)(194,230)(195,231)(196,232)(197,233)(209,241)(210,242)(211,243)(212,244)(218,247)(235,248)(236,249)(237,250)(238,251)(240,253)(246,255)(254,256);(1,4)(2,12)(3,18)(5,24)(6,25)(7,26)(8,27)(9,23)(10,39)(11,44)(13,49)(14,50)(15,51)(16,52)(17,59)(19,64)(20,65)(21,66)(22,67)(28,74)(29,78)(30,79)(31,80)(32,75)(33,81)(34,82)(35,76)(36,83)(37,77)(38,94)(40,99)(41,100)(42,101)(43,102)(45,109)(46,110)(47,111)(48,112)(53,119)(54,120)(55,121)(56,122)(57,123)(58,124)(60,129)(61,130)(62,131)(63,132)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(84,149)(85,150)(86,151)(87,155)(88,156)(89,157)(90,152)(91,153)(92,158)(93,154)(95,164)(96,165)(97,166)(98,167)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(125,194)(126,195)(127,196)(128,197)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(145,209)(146,210)(147,211)(148,212)(159,214)(160,215)(161,216)(162,218)(163,217)(168,220)(169,221)(170,222)(171,223)(172,224)(173,225)(180,230)(181,231)(182,232)(183,233)(190,235)(191,236)(192,237)(193,238)(198,240)(205,241)(206,242)(207,243)(208,244)(213,246)(219,247)(226,248)(227,249)(228,250)(229,251)(234,253)(239,254)(245,255)(252,256);(5,28)(13,40)(19,60)(24,74)(29,84)(30,85)(31,86)(45,95)(49,99)(53,103)(54,104)(55,105)(64,129)(68,133)(69,134)(70,135)(78,149)(79,150)(80,151)(87,159)(88,160)(89,161)(109,164)(113,168)(114,169)(115,170)(119,174)(120,175)(121,176)(125,180)(126,181)(127,182)(139,199)(140,200)(141,201)(145,205)(146,206)(147,207)(155,214)(156,215)(157,216)(162,219)(184,220)(185,221)(186,222)(190,226)(191,227)(192,228)(194,230)(195,231)(196,232)(198,234)(209,241)(210,242)(211,243)(213,245)(218,247)(235,248)(236,249)(237,250)(239,252)(240,253)(246,255)(254,256);(1,5)(2,13)(3,19)(4,24)(6,29)(7,30)(8,31)(9,28)(10,40)(11,45)(12,49)(14,53)(15,54)(16,55)(17,60)(18,64)(20,68)(21,69)(22,70)(23,74)(25,78)(26,79)(27,80)(32,84)(33,87)(34,88)(35,85)(36,89)(37,86)(38,95)(39,99)(41,103)(42,104)(43,105)(44,109)(46,113)(47,114)(48,115)(50,119)(51,120)(52,121)(56,125)(57,126)(58,127)(59,129)(61,133)(62,134)(63,135)(65,139)(66,140)(67,141)(71,145)(72,146)(73,147)(75,149)(76,150)(77,151)(81,155)(82,156)(83,157)(90,159)(91,160)(92,162)(93,161)(94,164)(96,168)(97,169)(98,170)(100,174)(101,175)(102,176)(106,180)(107,181)(108,182)(110,184)(111,185)(112,186)(116,190)(117,191)(118,192)(122,194)(123,195)(124,196)(128,198)(130,199)(131,200)(132,201)(136,205)(137,206)(138,207)(142,209)(143,210)(144,211)(148,213)(152,214)(153,215)(154,216)(158,218)(163,219)(165,220)(166,221)(167,222)(171,226)(172,227)(173,228)(177,230)(178,231)(179,232)(183,234)(187,235)(188,236)(189,237)(193,239)(197,240)(202,241)(203,242)(204,243)(208,245)(212,246)(217,247)(223,248)(224,249)(225,250)(229,252)(233,253)(238,254)(244,255)(251,256);(6,32)(14,41)(20,61)(25,75)(29,84)(33,90)(34,91)(46,96)(50,100)(53,103)(56,106)(57,107)(65,130)(68,133)(71,136)(72,137)(78,149)(81,152)(82,153)(87,159)(88,160)(92,163)(110,165)(113,168)(116,171)(117,172)(119,174)(122,177)(123,178)(125,180)(126,181)(128,183)(139,199)(142,202)(143,203)(145,205)(146,206)(148,208)(155,214)(156,215)(158,217)(162,219)(184,220)(187,223)(188,224)(190,226)(191,227)(193,229)(194,230)(195,231)(197,233)(198,234)(209,241)(210,242)(212,244)(213,245)(218,247)(235,248)(236,249)(238,251)(239,252)(240,253)(246,255)(254,256);(1,6)(2,14)(3,20)(4,25)(5,29)(7,33)(8,34)(9,32)(10,41)(11,46)(12,50)(13,53)(15,56)(16,57)(17,61)(18,65)(19,68)(21,71)(22,72)(23,75)(24,78)(26,81)(27,82)(28,84)(30,87)(31,88)(35,90)(36,92)(37,91)(38,96)(39,100)(40,103)(42,106)(43,107)(44,110)(45,113)(47,116)(48,117)(49,119)(51,122)(52,123)(54,125)(55,126)(58,128)(59,130)(60,133)(62,136)(63,137)(64,139)(66,142)(67,143)(69,145)(70,146)(73,148)(74,149)(76,152)(77,153)(79,155)(80,156)(83,158)(85,159)(86,160)(89,162)(93,163)(94,165)(95,168)(97,171)(98,172)(99,174)(101,177)(102,178)(104,180)(105,181)(108,183)(109,184)(111,187)(112,188)(114,190)(115,191)(118,193)(120,194)(121,195)(124,197)(127,198)(129,199)(131,202)(132,203)(134,205)(135,206)(138,208)(140,209)(141,210)(144,212)(147,213)(150,214)(151,215)(154,217)(157,218)(161,219)(164,220)(166,223)(167,224)(169,226)(170,227)(173,229)(175,230)(176,231)(179,233)(182,234)(185,235)(186,236)(189,238)(192,239)(196,240)(200,241)(201,242)(204,244)(207,245)(211,246)(216,247)(221,248)(222,249)(225,251)(228,252)(232,253)(237,254)(243,255)(250,256);(7,35)(15,42)(21,62)(26,76)(30,85)(33,90)(36,93)(47,97)(51,101)(54,104)(56,106)(58,108)(66,131)(69,134)(71,136)(73,138)(79,150)(81,152)(83,154)(87,159)(89,161)(92,163)(111,166)(114,169)(116,171)(118,173)(120,175)(122,177)(124,179)(125,180)(127,182)(128,183)(140,200)(142,202)(144,204)(145,205)(147,207)(148,208)(155,214)(157,216)(158,217)(162,219)(185,221)(187,223)(189,225)(190,226)(192,228)(193,229)(194,230)(196,232)(197,233)(198,234)(209,241)(211,243)(212,244)(213,245)(218,247)(235,248)(237,250)(238,251)(239,252)(240,253)(246,255)(254,256);(1,7)(2,15)(3,21)(4,26)(5,30)(6,33)(8,36)(9,35)(10,42)(11,47)(12,51)(13,54)(14,56)(16,58)(17,62)(18,66)(19,69)(20,71)(22,73)(23,76)(24,79)(25,81)(27,83)(28,85)(29,87)(31,89)(32,90)(34,92)(37,93)(38,97)(39,101)(40,104)(41,106)(43,108)(44,111)(45,114)(46,116)(48,118)(49,120)(50,122)(52,124)(53,125)(55,127)(57,128)(59,131)(60,134)(61,136)(63,138)(64,140)(65,142)(67,144)(68,145)(70,147)(72,148)(74,150)(75,152)(77,154)(78,155)(80,157)(82,158)(84,159)(86,161)(88,162)(91,163)(94,166)(95,169)(96,171)(98,173)(99,175)(100,177)(102,179)(103,180)(105,182)(107,183)(109,185)(110,187)(112,189)(113,190)(115,192)(117,193)(119,194)(121,196)(123,197)(126,198)(129,200)(130,202)(132,204)(133,205)(135,207)(137,208)(139,209)(141,211)(143,212)(146,213)(149,214)(151,216)(153,217)(156,218)(160,219)(164,221)(165,223)(167,225)(168,226)(170,228)(172,229)(174,230)(176,232)(178,233)(181,234)(184,235)(186,237)(188,238)(191,239)(195,240)(199,241)(201,243)(203,244)(206,245)(210,246)(215,247)(220,248)(222,250)(224,251)(227,252)(231,253)(236,254)(242,255)(249,256);(8,37)(16,43)(22,63)(27,77)(31,86)(34,91)(36,93)(48,98)(52,102)(55,105)(57,107)(58,108)(67,132)(70,135)(72,137)(73,138)(80,151)(82,153)(83,154)(88,160)(89,161)(92,163)(112,167)(115,170)(117,172)(118,173)(121,176)(123,178)(124,179)(126,181)(127,182)(128,183)(141,201)(143,203)(144,204)(146,206)(147,207)(148,208)(156,215)(157,216)(158,217)(162,219)(186,222)(188,224)(189,225)(191,227)(192,228)(193,229)(195,231)(196,232)(197,233)(198,234)(210,242)(211,243)(212,244)(213,245)(218,247)(236,249)(237,250)(238,251)(239,252)(240,253)(246,255)(254,256);(1,8)(2,16)(3,22)(4,27)(5,31)(6,34)(7,36)(9,37)(10,43)(11,48)(12,52)(13,55)(14,57)(15,58)(17,63)(18,67)(19,70)(20,72)(21,73)(23,77)(24,80)(25,82)(26,83)(28,86)(29,88)(30,89)(32,91)(33,92)(35,93)(38,98)(39,102)(40,105)(41,107)(42,108)(44,112)(45,115)(46,117)(47,118)(49,121)(50,123)(51,124)(53,126)(54,127)(56,128)(59,132)(60,135)(61,137)(62,138)(64,141)(65,143)(66,144)(68,146)(69,147)(71,148)(74,151)(75,153)(76,154)(78,156)(79,157)(81,158)(84,160)(85,161)(87,162)(90,163)(94,167)(95,170)(96,172)(97,173)(99,176)(100,178)(101,179)(103,181)(104,182)(106,183)(109,186)(110,188)(111,189)(113,191)(114,192)(116,193)(119,195)(120,196)(122,197)(125,198)(129,201)(130,203)(131,204)(133,206)(134,207)(136,208)(139,210)(140,211)(142,212)(145,213)(149,215)(150,216)(152,217)(155,218)(159,219)(164,222)(165,224)(166,225)(168,227)(169,228)(171,229)(174,231)(175,232)(177,233)(180,234)(184,236)(185,237)(187,238)(190,239)(194,240)(199,242)(200,243)(202,244)(205,245)(209,246)(214,247)(220,249)(221,250)(223,251)(226,252)(230,253)(235,254)(241,255)(248,256)Polycyclic presentation:
2^15|b^a=bo,d^c=do,f^e=fo,h^g=ho,j^i=jo,l^k=lo,n^m=no16385 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.15);;
gap> SetConjugate(coll, 4, 3, F.4*F.15);;
gap> SetConjugate(coll, 6, 5, F.6*F.15);;
gap> SetConjugate(coll, 8, 7, F.8*F.15);;
gap> SetConjugate(coll, 10, 9, F.10*F.15);;
gap> SetConjugate(coll, 12, 11, F.12*F.15);;
gap> SetConjugate(coll, 14, 13, F.14*F.15);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);