Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #513316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 256 points:
(1,2,9,16)(3,17,23,75)(4,11,24,53)(5,29,35,71)(6,36,40,127)(7,41,44,133)(8,45,43,15)(10,47,52,19)(12,58,62,32)(13,63,66,31)(14,67,69,38)(18,72,76,179)(20,83,85,192)(21,86,87,194)(22,88,46,74)(25,93,89,176)(26,99,90,198)(27,103,91,201)(28,105,92,57)(30,64,113,173)(33,119,122,182)(34,123,121,109)(37,59,130,165)(39,131,115,126)(42,73,134,183)(48,137,139,77)(49,143,145,111)(50,146,147,110)(51,82,70,136)(54,151,148,96)(55,155,149,95)(56,158,150,101)(60,166,168,225)(61,118,135,163)(65,116,125,170)(68,138,175,81)(78,188,184,252)(79,190,185,253)(80,191,106,181)(84,193,128,167)(94,156,195,235)(97,213,196,251)(98,215,197,205)(100,152,199,233)(102,221,200,218)(104,180,202,250)(107,162,177,114)(108,161,178,117)(112,174,124,172)(120,171,226,144)(129,169,132,164)(140,230,228,207)(141,232,229,206)(142,187,160,224)(153,241,234,256)(154,212,223,239)(157,210,217,244)(159,227,236,186)(189,254,219,242)(203,238,248,209)(204,237,249,211)(208,247,216,246)(214,245,255,231)(220,243,222,240);(1,3,9,23)(2,10,16,52)(4,18,24,76)(5,30,35,113)(6,37,40,130)(7,42,44,134)(8,46,43,22)(11,48,53,139)(12,59,62,165)(13,64,66,173)(14,68,69,175)(15,70,45,51)(17,71,75,29)(19,58,47,32)(20,63,85,31)(21,67,87,38)(25,94,89,195)(26,100,90,199)(27,104,91,202)(28,106,92,80)(33,120,122,226)(34,124,121,112)(36,49,127,145)(39,132,115,129)(41,50,133,147)(54,152,148,233)(55,156,149,235)(56,159,150,236)(57,160,105,142)(60,167,168,193)(61,169,135,164)(65,174,125,172)(72,176,179,93)(73,182,183,119)(74,109,88,123)(77,151,137,96)(78,155,184,95)(79,158,185,101)(81,166,138,225)(82,118,136,163)(83,177,192,107)(84,116,128,170)(86,178,194,108)(97,214,196,255)(98,216,197,208)(99,140,198,228)(102,222,200,220)(103,141,201,229)(110,162,146,114)(111,161,143,117)(126,171,131,144)(153,242,234,254)(154,243,223,240)(157,247,217,246)(180,251,250,213)(181,205,191,215)(186,241,227,256)(187,212,224,239)(188,248,252,203)(189,210,219,244)(190,249,253,204)(206,238,232,209)(207,237,230,211)(218,245,221,231);(1,4)(2,11)(3,18)(5,25)(6,26)(7,27)(8,28)(9,24)(10,48)(12,54)(13,55)(14,56)(15,57)(16,53)(17,72)(19,77)(20,78)(21,79)(22,80)(23,76)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,89)(36,99)(37,100)(38,101)(39,102)(40,90)(41,103)(42,104)(43,92)(44,91)(45,105)(46,106)(47,137)(49,140)(50,141)(51,142)(52,139)(58,151)(59,152)(60,153)(61,154)(62,148)(63,155)(64,156)(65,157)(66,149)(67,158)(68,159)(69,150)(70,160)(71,176)(73,180)(74,181)(75,179)(81,186)(82,187)(83,188)(84,189)(85,184)(86,190)(87,185)(88,191)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(113,195)(114,209)(115,200)(116,210)(117,211)(118,212)(119,213)(120,214)(121,197)(122,196)(123,215)(124,216)(125,217)(126,218)(127,198)(128,219)(129,220)(130,199)(131,221)(132,222)(133,201)(134,202)(135,223)(136,224)(138,227)(143,230)(144,231)(145,228)(146,232)(147,229)(161,237)(162,238)(163,239)(164,240)(165,233)(166,241)(167,242)(168,234)(169,243)(170,244)(171,245)(172,246)(173,235)(174,247)(175,236)(177,248)(178,249)(182,251)(183,250)(192,252)(193,254)(194,253)(225,256)(226,255);(4,24)(11,53)(18,76)(25,89)(26,90)(27,91)(28,92)(48,139)(54,148)(55,149)(56,150)(57,105)(72,179)(77,137)(78,184)(79,185)(80,106)(93,176)(94,195)(95,155)(96,151)(97,196)(98,197)(99,198)(100,199)(101,158)(102,200)(103,201)(104,202)(140,228)(141,229)(142,160)(152,233)(153,234)(154,223)(156,235)(157,217)(159,236)(180,250)(181,191)(186,227)(187,224)(188,252)(189,219)(190,253)(203,248)(204,249)(205,215)(206,232)(207,230)(208,216)(209,238)(210,244)(211,237)(212,239)(213,251)(214,255)(218,221)(220,222)(231,245)(240,243)(241,256)(242,254)(246,247)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=be,e^a=ef,f^a=fg,g^a=gh,h^a=hi,e^b=ef,f^b=fg,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420514);