Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D64 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| socle | C2 | #7537 |
| prime radical | #13741 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ 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:
(1,2,5,12,9,17,37,16)(3,18,20,77,25,80,24,97)(4,26,29,98,33,103,116,102)(6,38,34,126,42,136,128,135)(7,43,35,129,45,146,131,145)(8,46,36,15,48,64,132,75)(10,49,50,84,54,83,76,19)(11,55,57,119,61,164,60,30)(13,65,62,113,69,182,68,28)(14,70,63,141,73,190,72,40)(21,88,85,203,91,206,90,209)(22,92,86,205,94,211,93,212)(23,95,87,198,96,213,47,79)(27,66,104,178,109,185,108,189)(31,120,114,230,122,236,232,235)(32,123,115,101,125,216,233,221)(39,56,127,165,140,167,139,177)(41,142,111,134,144,239,229,241)(44,78,130,196,148,197,147,210)(51,155,153,226,157,225,195,106)(52,158,154,224,159,223,191,105)(53,82,74,152,160,202,194,150)(58,170,168,248,173,249,172,199)(59,174,169,246,176,118,149,163)(67,186,179,253,188,112,133,181)(71,151,180,201,192,200,193,81)(89,207,204,250,208,251,137,171)(99,162,214,228,218,245,217,110)(100,161,215,234,220,244,219,117)(107,187,222,254,227,256,124,184)(121,183,231,243,238,255,237,156)(138,175,240,247,242,252,143,166);(1,3)(2,10)(4,27)(5,20)(6,39)(7,44)(8,47)(9,25)(11,56)(12,50)(13,66)(14,71)(15,74)(16,76)(17,54)(18,26)(19,30)(21,28)(22,40)(23,36)(24,37)(29,104)(31,121)(32,124)(33,109)(34,127)(35,130)(38,51)(41,143)(42,140)(43,52)(45,148)(46,53)(48,87)(49,55)(57,165)(58,171)(59,175)(60,177)(61,167)(62,178)(63,180)(64,160)(65,85)(67,187)(68,189)(69,185)(70,86)(72,193)(73,192)(75,194)(77,98)(78,120)(79,123)(80,103)(81,199)(82,163)(83,164)(84,119)(88,99)(89,181)(90,182)(91,113)(92,100)(93,190)(94,141)(95,101)(96,132)(97,102)(105,110)(106,117)(107,115)(108,116)(111,138)(112,137)(114,231)(118,150)(122,238)(125,222)(126,153)(128,139)(129,154)(131,147)(133,184)(134,183)(135,195)(136,157)(142,156)(144,240)(145,191)(146,159)(149,166)(151,170)(152,174)(155,161)(158,162)(168,207)(169,247)(172,251)(173,250)(176,252)(179,254)(186,204)(188,256)(196,230)(197,236)(198,216)(200,249)(201,248)(202,246)(203,214)(205,215)(206,218)(208,253)(209,217)(210,235)(211,220)(212,219)(213,221)(223,245)(224,228)(225,244)(226,234)(227,233)(229,242)(232,237)(239,243)(241,255)Polycyclic presentation:
2^9|a^2=d,c^2=ef,d^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=ce,e^b=ef,f^b=fg,g^b=gh,h^b=hi134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2030);