Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21698 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,6)(7,8,10,9)(11,12,16,20)(13,21,25,35)(14,18,26,32)(15,28,22,19)(17,29,31,24)(23,34,37,42)(27,39,36,33)(30,40,41,38);(11,13,22,29,16,25,15,24)(12,17,28,35,20,31,19,21)(14,23,36,40,26,37,27,38)(18,30,39,42,32,41,33,34);(3,4,5,6)(11,14,16,26)(12,18,20,32)(13,23,25,37)(15,27,22,36)(17,30,31,41)(19,33,28,39)(21,34,35,42)(24,38,29,40);(7,9,10,8)(14,26)(18,32)(23,37)(27,36)(30,41)(33,39)(34,42)(38,40)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280935);