Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C22×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,11,13,12,14,15,16);(1,2)(3,8)(4,6)(5,7)(17,18,20,19,21,22,24,23);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(17,19,20,22,21,23,24,18)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91575);