2₋¹⁺¹² #72274295

order	8192
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹² · Schur multiplier (H₂) = C₂⁶⁵

Aliases

extraspecial

2₋¹⁺¹²

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂	C₂¹²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ G
LECS	G ▷ C₂ ▷ C₁

488194502149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 256
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 12 generators on 256 points:

(1,2,9,16)(3,17,23,10)(4,11,24,44)(5,12,29,49)(6,13,33,53)(7,14,36,56)(8,15,38,58)(18,59,64,39)(19,60,69,40)(20,61,73,41)(21,62,76,42)(22,63,78,43)(25,45,79,109)(26,46,80,110)(27,47,81,111)(28,48,82,112)(30,50,89,119)(31,51,90,120)(32,52,91,121)(34,54,95,125)(35,55,96,126)(37,57,98,128)(65,129,139,99)(66,130,140,100)(67,131,141,101)(68,132,142,102)(70,133,149,103)(71,134,150,104)(72,135,151,105)(74,136,155,106)(75,137,156,107)(77,138,158,108)(83,113,159,184)(84,114,160,185)(85,115,161,186)(86,116,162,187)(87,117,163,188)(88,118,164,189)(92,122,169,194)(93,123,170,195)(94,124,171,196)(97,127,173,198)(143,199,209,174)(144,200,210,175)(145,201,211,176)(146,202,212,177)(147,203,213,178)(148,204,214,179)(152,205,219,180)(153,206,220,181)(154,207,221,182)(157,208,223,183)(165,190,224,235)(166,191,225,236)(167,192,226,237)(168,193,227,238)(172,197,229,240)(215,241,246,230)(216,242,247,231)(217,243,248,232)(218,244,249,233)(222,245,251,234)(228,239,252,254)(250,255,256,253);(1,3,9,23)(2,10,16,17)(4,18,24,64)(5,19,29,69)(6,20,33,73)(7,21,36,76)(8,22,38,78)(11,39,44,59)(12,40,49,60)(13,41,53,61)(14,42,56,62)(15,43,58,63)(25,65,79,139)(26,66,80,140)(27,67,81,141)(28,68,82,142)(30,70,89,149)(31,71,90,150)(32,72,91,151)(34,74,95,155)(35,75,96,156)(37,77,98,158)(45,99,109,129)(46,100,110,130)(47,101,111,131)(48,102,112,132)(50,103,119,133)(51,104,120,134)(52,105,121,135)(54,106,125,136)(55,107,126,137)(57,108,128,138)(83,143,159,209)(84,144,160,210)(85,145,161,211)(86,146,162,212)(87,147,163,213)(88,148,164,214)(92,152,169,219)(93,153,170,220)(94,154,171,221)(97,157,173,223)(113,174,184,199)(114,175,185,200)(115,176,186,201)(116,177,187,202)(117,178,188,203)(118,179,189,204)(122,180,194,205)(123,181,195,206)(124,182,196,207)(127,183,198,208)(165,215,224,246)(166,216,225,247)(167,217,226,248)(168,218,227,249)(172,222,229,251)(190,230,235,241)(191,231,236,242)(192,232,237,243)(193,233,238,244)(197,234,240,245)(228,250,252,256)(239,253,254,255);(4,24)(11,44)(18,64)(25,79)(26,80)(27,81)(28,82)(39,59)(45,109)(46,110)(47,111)(48,112)(65,139)(66,140)(67,141)(68,142)(83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(99,129)(100,130)(101,131)(102,132)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(143,209)(144,210)(145,211)(146,212)(147,213)(148,214)(165,224)(166,225)(167,226)(168,227)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(190,235)(191,236)(192,237)(193,238)(215,246)(216,247)(217,248)(218,249)(228,252)(230,241)(231,242)(232,243)(233,244)(239,254)(250,256)(253,255);(1,4)(2,11)(3,18)(5,25)(6,26)(7,27)(8,28)(9,24)(10,39)(12,45)(13,46)(14,47)(15,48)(16,44)(17,59)(19,65)(20,66)(21,67)(22,68)(23,64)(29,79)(30,83)(31,84)(32,85)(33,80)(34,86)(35,87)(36,81)(37,88)(38,82)(40,99)(41,100)(42,101)(43,102)(49,109)(50,113)(51,114)(52,115)(53,110)(54,116)(55,117)(56,111)(57,118)(58,112)(60,129)(61,130)(62,131)(63,132)(69,139)(70,143)(71,144)(72,145)(73,140)(74,146)(75,147)(76,141)(77,148)(78,142)(89,159)(90,160)(91,161)(92,165)(93,166)(94,167)(95,162)(96,163)(97,168)(98,164)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(119,184)(120,185)(121,186)(122,190)(123,191)(124,192)(125,187)(126,188)(127,193)(128,189)(133,199)(134,200)(135,201)(136,202)(137,203)(138,204)(149,209)(150,210)(151,211)(152,215)(153,216)(154,217)(155,212)(156,213)(157,218)(158,214)(169,224)(170,225)(171,226)(172,228)(173,227)(180,230)(181,231)(182,232)(183,233)(194,235)(195,236)(196,237)(197,239)(198,238)(205,241)(206,242)(207,243)(208,244)(219,246)(220,247)(221,248)(222,250)(223,249)(229,252)(234,253)(240,254)(245,255)(251,256);(5,29)(12,49)(19,69)(25,79)(30,89)(31,90)(32,91)(40,60)(45,109)(50,119)(51,120)(52,121)(65,139)(70,149)(71,150)(72,151)(83,159)(84,160)(85,161)(92,169)(93,170)(94,171)(99,129)(103,133)(104,134)(105,135)(113,184)(114,185)(115,186)(122,194)(123,195)(124,196)(143,209)(144,210)(145,211)(152,219)(153,220)(154,221)(165,224)(166,225)(167,226)(172,229)(174,199)(175,200)(176,201)(180,205)(181,206)(182,207)(190,235)(191,236)(192,237)(197,240)(215,246)(216,247)(217,248)(222,251)(228,252)(230,241)(231,242)(232,243)(234,245)(239,254)(250,256)(253,255);(1,5)(2,12)(3,19)(4,25)(6,30)(7,31)(8,32)(9,29)(10,40)(11,45)(13,50)(14,51)(15,52)(16,49)(17,60)(18,65)(20,70)(21,71)(22,72)(23,69)(24,79)(26,83)(27,84)(28,85)(33,89)(34,92)(35,93)(36,90)(37,94)(38,91)(39,99)(41,103)(42,104)(43,105)(44,109)(46,113)(47,114)(48,115)(53,119)(54,122)(55,123)(56,120)(57,124)(58,121)(59,129)(61,133)(62,134)(63,135)(64,139)(66,143)(67,144)(68,145)(73,149)(74,152)(75,153)(76,150)(77,154)(78,151)(80,159)(81,160)(82,161)(86,165)(87,166)(88,167)(95,169)(96,170)(97,172)(98,171)(100,174)(101,175)(102,176)(106,180)(107,181)(108,182)(110,184)(111,185)(112,186)(116,190)(117,191)(118,192)(125,194)(126,195)(127,197)(128,196)(130,199)(131,200)(132,201)(136,205)(137,206)(138,207)(140,209)(141,210)(142,211)(146,215)(147,216)(148,217)(155,219)(156,220)(157,222)(158,221)(162,224)(163,225)(164,226)(168,228)(173,229)(177,230)(178,231)(179,232)(183,234)(187,235)(188,236)(189,237)(193,239)(198,240)(202,241)(203,242)(204,243)(208,245)(212,246)(213,247)(214,248)(218,250)(223,251)(227,252)(233,253)(238,254)(244,255)(249,256);(6,33)(13,53)(20,73)(26,80)(30,89)(34,95)(35,96)(41,61)(46,110)(50,119)(54,125)(55,126)(66,140)(70,149)(74,155)(75,156)(83,159)(86,162)(87,163)(92,169)(93,170)(97,173)(100,130)(103,133)(106,136)(107,137)(113,184)(116,187)(117,188)(122,194)(123,195)(127,198)(143,209)(146,212)(147,213)(152,219)(153,220)(157,223)(165,224)(166,225)(168,227)(172,229)(174,199)(177,202)(178,203)(180,205)(181,206)(183,208)(190,235)(191,236)(193,238)(197,240)(215,246)(216,247)(218,249)(222,251)(228,252)(230,241)(231,242)(233,244)(234,245)(239,254)(250,256)(253,255);(1,6)(2,13)(3,20)(4,26)(5,30)(7,34)(8,35)(9,33)(10,41)(11,46)(12,50)(14,54)(15,55)(16,53)(17,61)(18,66)(19,70)(21,74)(22,75)(23,73)(24,80)(25,83)(27,86)(28,87)(29,89)(31,92)(32,93)(36,95)(37,97)(38,96)(39,100)(40,103)(42,106)(43,107)(44,110)(45,113)(47,116)(48,117)(49,119)(51,122)(52,123)(56,125)(57,127)(58,126)(59,130)(60,133)(62,136)(63,137)(64,140)(65,143)(67,146)(68,147)(69,149)(71,152)(72,153)(76,155)(77,157)(78,156)(79,159)(81,162)(82,163)(84,165)(85,166)(88,168)(90,169)(91,170)(94,172)(98,173)(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)(128,198)(129,199)(131,202)(132,203)(134,205)(135,206)(138,208)(139,209)(141,212)(142,213)(144,215)(145,216)(148,218)(150,219)(151,220)(154,222)(158,223)(160,224)(161,225)(164,227)(167,228)(171,229)(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)(210,246)(211,247)(214,249)(217,250)(221,251)(226,252)(232,253)(237,254)(243,255)(248,256);(7,36)(14,56)(21,76)(27,81)(31,90)(34,95)(37,98)(42,62)(47,111)(51,120)(54,125)(57,128)(67,141)(71,150)(74,155)(77,158)(84,160)(86,162)(88,164)(92,169)(94,171)(97,173)(101,131)(104,134)(106,136)(108,138)(114,185)(116,187)(118,189)(122,194)(124,196)(127,198)(144,210)(146,212)(148,214)(152,219)(154,221)(157,223)(165,224)(167,226)(168,227)(172,229)(175,200)(177,202)(179,204)(180,205)(182,207)(183,208)(190,235)(192,237)(193,238)(197,240)(215,246)(217,248)(218,249)(222,251)(228,252)(230,241)(232,243)(233,244)(234,245)(239,254)(250,256)(253,255);(1,7)(2,14)(3,21)(4,27)(5,31)(6,34)(8,37)(9,36)(10,42)(11,47)(12,51)(13,54)(15,57)(16,56)(17,62)(18,67)(19,71)(20,74)(22,77)(23,76)(24,81)(25,84)(26,86)(28,88)(29,90)(30,92)(32,94)(33,95)(35,97)(38,98)(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)(58,128)(59,131)(60,134)(61,136)(63,138)(64,141)(65,144)(66,146)(68,148)(69,150)(70,152)(72,154)(73,155)(75,157)(78,158)(79,160)(80,162)(82,164)(83,165)(85,167)(87,168)(89,169)(91,171)(93,172)(96,173)(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)(139,210)(140,212)(142,214)(143,215)(145,217)(147,218)(149,219)(151,221)(153,222)(156,223)(159,224)(161,226)(163,227)(166,228)(170,229)(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)(209,246)(211,248)(213,249)(216,250)(220,251)(225,252)(231,253)(236,254)(242,255)(247,256);(8,38)(15,58)(22,78)(28,82)(32,91)(35,96)(37,98)(43,63)(48,112)(52,121)(55,126)(57,128)(68,142)(72,151)(75,156)(77,158)(85,161)(87,163)(88,164)(93,170)(94,171)(97,173)(102,132)(105,135)(107,137)(108,138)(115,186)(117,188)(118,189)(123,195)(124,196)(127,198)(145,211)(147,213)(148,214)(153,220)(154,221)(157,223)(166,225)(167,226)(168,227)(172,229)(176,201)(178,203)(179,204)(181,206)(182,207)(183,208)(191,236)(192,237)(193,238)(197,240)(216,247)(217,248)(218,249)(222,251)(228,252)(231,242)(232,243)(233,244)(234,245)(239,254)(250,256)(253,255);(1,8)(2,15)(3,22)(4,28)(5,32)(6,35)(7,37)(9,38)(10,43)(11,48)(12,52)(13,55)(14,57)(16,58)(17,63)(18,68)(19,72)(20,75)(21,77)(23,78)(24,82)(25,85)(26,87)(27,88)(29,91)(30,93)(31,94)(33,96)(34,97)(36,98)(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)(64,142)(65,145)(66,147)(67,148)(69,151)(70,153)(71,154)(73,156)(74,157)(76,158)(79,161)(80,163)(81,164)(83,166)(84,167)(86,168)(89,170)(90,171)(92,172)(95,173)(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)(139,211)(140,213)(141,214)(143,216)(144,217)(146,218)(149,220)(150,221)(152,222)(155,223)(159,225)(160,226)(162,227)(165,228)(169,229)(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)(209,247)(210,248)(212,249)(215,250)(219,251)(224,252)(230,253)(235,254)(241,255)(246,256)

Polycyclic presentation:

2^13|a^2=m,b^2=m,b^a=bm,d^c=dm,f^e=fm,h^g=hm,j^i=jm,l^k=lm

Conjugacy classes & character table

4097 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72290098, C₂×2₋¹⁺¹²
maximal normal subgroup of	#72290098, C₂×2₋¹⁺¹²
maximal quotient of	C₂×2₋¹⁺¹²

GAP

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.13);;
gap> SetPower(coll, 2, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 8, 7, F.8*F.13);;
gap> SetConjugate(coll, 10, 9, F.10*F.13);;
gap> SetConjugate(coll, 12, 11, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

2−1+12 #72274295