Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#2222 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2);(3,4);(5,6,12,18)(7,19,25,47)(8,26,29,43)(9,30,32,41)(10,17,11,16)(13,34,38,22)(14,33,40,28)(15,31,42,20)(21,50,53,67)(23,46,24,45)(27,58,59,39)(35,62,63,68)(36,54,37,48)(44,55,66,52)(49,57,60,65)(51,56,61,64);(5,7,20,48,11,24,32,34,12,25,31,54,10,23,9,22)(6,13,30,46,17,37,42,47,18,38,41,45,16,36,15,19)(8,21,49,68,28,52,59,61,29,53,60,62,33,55,27,51)(14,35,57,50,26,56,58,66,40,63,65,67,43,64,39,44);(5,8)(6,14)(7,21)(9,27)(10,33)(11,28)(12,29)(13,35)(15,39)(16,43)(17,26)(18,40)(19,44)(20,49)(22,51)(23,55)(24,52)(25,53)(30,57)(31,60)(32,59)(34,61)(36,64)(37,56)(38,63)(41,65)(42,58)(45,67)(46,50)(47,66)(48,68)(54,62)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530182);