Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#2244 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C22×D8 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(14,20)(15,26)(16,21)(17,28)(18,22)(23,24)(25,27);(6,9)(10,12)(13,14,21,15,18,24,28,27,19,25,17,23,22,26,16,20);(5,7,8,11)(6,10,9,12)(13,15,16,14,22,20,17,26,19,23,28,25,18,27,21,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403238);