Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | C2×D32 |
| prime radical | C23 | D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2,3,4)(5,6)(7,20)(8,15)(9,31)(10,36)(11,40)(12,42)(13,19)(14,23)(16,21)(17,32)(18,37)(22,53)(24,71)(25,75)(26,77)(27,57)(28,78)(29,81)(30,52)(33,87)(34,89)(35,84)(38,92)(39,91)(41,94)(43,64)(44,98)(45,100)(46,70)(47,59)(48,79)(49,102)(50,63)(51,86)(54,58)(55,67)(56,72)(60,83)(61,82)(62,96)(65,117)(66,109)(68,93)(69,95)(73,123)(74,122)(76,124)(80,126)(85,90)(88,128)(97,130)(99,132)(101,116)(103,113)(104,110)(105,115)(106,131)(107,112)(108,119)(111,125)(114,127)(118,121)(120,129);(5,7,21,58,10,24,60,100,37,72,102,131,85,118,34,69,12,26,62,111,38,73,88,120,41,76,114,132,86,119,35,70,13,27,63,112,39,74,80,116,30,66,113,130,79,115,28,64,8,22,59,110,29,65,33,68,11,25,61,98,32,67,9,23)(6,14,31,55,17,44,82,75,40,93,87,117,81,104,47,53,15,43,78,105,48,97,103,109,52,101,126,122,91,107,50,57,19,46,84,108,51,99,127,124,94,129,128,123,92,125,96,77,42,95,89,121,90,106,49,56,18,45,83,71,36,54,16,20);(5,8,13,12)(6,15,19,42)(7,22,27,26)(9,28,35,34)(10,29,39,38)(11,30,41,37)(14,43,46,95)(16,47,50,96)(17,48,51,90)(18,40,52,94)(20,53,57,77)(21,59,63,62)(23,64,70,69)(24,65,74,73)(25,66,76,72)(31,78,84,89)(32,79,86,85)(33,80,88,60)(36,81,91,92)(44,97,99,106)(45,93,101,129)(49,82,103,127)(54,104,107,125)(55,105,108,121)(56,75,109,124)(58,110,112,111)(61,113,114,102)(67,115,119,118)(68,116,120,100)(71,117,122,123)(83,87,126,128)(98,130,132,131)Polycyclic presentation:
2^9|a^2=e,b^2=df,c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60769);