Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C23×C42 |
| direct product | C8×#340 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,11,13,12,14,15,16);(1,2)(3,4)(5,6,7,8)(18,22)(19,20)(23,24);(2,4)(17,18,20,23,21,24,19,22)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12478);