Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×#491 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(9,10,11,13,12,14,15,16);(1,2)(3,6)(4,5)(7,8)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,27,22,26);(2,5)(6,8)(17,19,21,20,23,30,22,29)(18,24,26,25,28,32,27,31);(1,3,4,7)(2,6,5,8)(17,20,21,30,23,29,22,19)(18,25,26,32,28,31,27,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 119810);