Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#1295 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4);(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(18,21)(19,23);(10,13,15,14)(17,18,19,22,20,21,23,24)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72217670 |
| maximal normal subgroup of | #72217670 |
| maximal quotient of | #72213034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12637);