Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#2229 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C23 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4);(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,23,24,41)(17,29,27,42)(18,32,28,22)(20,36,34,25)(21,38,35,31)(26,43,33,39)(30,40,44,37);(5,7,8,12)(6,9,10,11)(14,19)(20,34)(21,35)(22,32)(23,41)(29,42)(37,40)(39,43);(13,15,16,24)(14,20,19,34)(17,30,27,44)(18,33,28,26)(21,39,35,43)(22,40,32,37)(23,42,41,29)(25,38,36,31)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403175);