Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C64 | D4 |
| commutator | C4 | C2×C64 |
| Frattini, cosocle | C2×C64 | C22 |
| socle | C2 | C22⋊C64 |
| prime radical | #41 | C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 256 points:
(1,2,5,13,6,14,32,58,7,15,33,59,37,63,102,142,8,16,34,60,38,64,103,143,41,67,106,146,111,151,190,219,9,17,35,61,39,65,104,144,42,68,107,147,112,152,191,220,44,70,109,149,114,154,193,222,116,156,195,224,197,226,244,252,10,18,36,62,40,66,105,145,43,69,108,148,113,153,192,221,45,71,110,150,115,155,194,223,117,157,196,225,198,227,245,253,4,12,27,53,28,54,92,133,29,55,93,134,95,136,180,212,30,56,94,135,96,137,181,213,97,138,182,214,183,215,239,250)(3,19,21,72,22,73,80,158,23,74,81,159,83,161,170,228,24,75,82,160,84,162,171,229,86,163,172,230,173,231,236,254,20,51,76,57,77,126,164,139,78,128,165,140,167,206,232,216,79,129,166,141,168,207,233,217,169,208,234,218,235,248,255,251,25,52,31,98,85,130,99,184,87,131,100,185,174,209,187,240,88,132,101,186,175,210,188,241,176,211,189,242,237,249,243,256,26,11,46,47,89,48,118,121,90,49,119,122,177,124,199,202,91,50,120,123,178,125,200,203,179,127,201,204,238,205,246,247);(1,3)(2,11)(4,26)(5,31)(6,22)(7,23)(8,24)(9,20)(10,25)(12,52)(13,57)(14,48)(15,49)(16,50)(17,19)(18,51)(21,36)(27,76)(28,89)(29,90)(30,91)(32,99)(33,100)(34,101)(35,46)(37,83)(38,84)(39,77)(40,85)(41,86)(42,78)(43,87)(44,79)(45,88)(47,62)(53,72)(54,130)(55,131)(56,132)(58,139)(59,140)(60,141)(61,98)(63,124)(64,125)(65,73)(66,126)(67,127)(68,74)(69,128)(70,75)(71,129)(80,105)(81,108)(82,110)(92,164)(93,165)(94,166)(95,177)(96,178)(97,179)(102,187)(103,188)(104,118)(106,189)(107,119)(109,120)(111,173)(112,167)(113,174)(114,168)(115,175)(116,169)(117,176)(121,145)(122,148)(123,150)(133,158)(134,159)(135,160)(136,209)(137,210)(138,211)(142,216)(143,217)(144,184)(146,218)(147,185)(149,186)(151,205)(152,161)(153,206)(154,162)(155,207)(156,163)(157,208)(170,192)(171,194)(172,196)(180,232)(181,233)(182,234)(183,238)(190,243)(191,199)(193,200)(195,201)(197,235)(198,237)(202,221)(203,223)(204,225)(212,228)(213,229)(214,230)(215,249)(219,251)(220,240)(222,241)(224,242)(226,231)(227,248)(236,245)(239,255)(244,246)(247,253)(250,254)(252,256)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^b=ci,d^b=di224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2014);