Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C64 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C128 | C22 |
| commutator, socle | C2 | C22×C64 |
| Frattini, cosocle | C64 | C23 |
| prime radical | #41 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C128 ◁ G | |
| LECS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 256 points:
(1,2,4,11,5,12,23,44,6,13,24,45,28,49,74,109,7,14,25,46,29,50,75,110,32,53,78,113,84,119,149,184,8,15,26,47,30,51,76,111,33,54,79,114,85,120,150,185,35,56,81,116,87,122,152,187,90,125,155,190,159,194,214,235,9,16,27,48,31,52,77,112,34,55,80,115,86,121,151,186,36,57,82,117,88,123,153,188,91,126,156,191,160,195,215,236,37,58,83,118,89,124,154,189,92,127,157,192,161,196,216,237,93,128,158,193,162,197,217,238,163,198,218,239,219,240,247,254)(3,10,18,39,19,40,64,99,20,41,65,100,68,103,139,174,21,42,66,101,69,104,140,175,71,106,142,177,145,180,209,230,22,43,67,102,70,105,141,176,72,107,143,178,146,181,210,231,73,108,144,179,147,182,211,232,148,183,212,233,213,234,246,253,17,38,59,94,60,95,129,164,61,96,130,165,133,168,199,220,62,97,131,166,134,169,200,221,136,171,202,223,205,226,241,248,63,98,132,167,135,170,201,222,137,172,203,224,206,227,242,249,138,173,204,225,207,228,243,250,208,229,244,251,245,252,255,256);(1,3)(2,10)(4,18)(5,19)(6,20)(7,21)(8,22)(9,17)(11,39)(12,40)(13,41)(14,42)(15,43)(16,38)(23,64)(24,65)(25,66)(26,67)(27,59)(28,68)(29,69)(30,70)(31,60)(32,71)(33,72)(34,61)(35,73)(36,62)(37,63)(44,99)(45,100)(46,101)(47,102)(48,94)(49,103)(50,104)(51,105)(52,95)(53,106)(54,107)(55,96)(56,108)(57,97)(58,98)(74,139)(75,140)(76,141)(77,129)(78,142)(79,143)(80,130)(81,144)(82,131)(83,132)(84,145)(85,146)(86,133)(87,147)(88,134)(89,135)(90,148)(91,136)(92,137)(93,138)(109,174)(110,175)(111,176)(112,164)(113,177)(114,178)(115,165)(116,179)(117,166)(118,167)(119,180)(120,181)(121,168)(122,182)(123,169)(124,170)(125,183)(126,171)(127,172)(128,173)(149,209)(150,210)(151,199)(152,211)(153,200)(154,201)(155,212)(156,202)(157,203)(158,204)(159,213)(160,205)(161,206)(162,207)(163,208)(184,230)(185,231)(186,220)(187,232)(188,221)(189,222)(190,233)(191,223)(192,224)(193,225)(194,234)(195,226)(196,227)(197,228)(198,229)(214,246)(215,241)(216,242)(217,243)(218,244)(219,245)(235,253)(236,248)(237,249)(238,250)(239,251)(240,252)(247,255)(254,256);(3,17)(10,38)(18,59)(19,60)(20,61)(21,62)(22,63)(39,94)(40,95)(41,96)(42,97)(43,98)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(174,220)(175,221)(176,222)(177,223)(178,224)(179,225)(180,226)(181,227)(182,228)(183,229)(209,241)(210,242)(211,243)(212,244)(213,245)(230,248)(231,249)(232,250)(233,251)(234,252)(246,255)(253,256)Polycyclic presentation:
2^9|a^2=d,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,c^b=ci320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60897);