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) = C24×C4 |
| direct product | C4×#2150 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C23×C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4);(6,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(5,6,7,8);(10,14)(15,20)(16,21)(22,24)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400290);