Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D64 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420505);