Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24849 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,11,14)(9,13,10,12)(15,16,19,29,20,25,21,24)(17,26,28,30,18,22,27,23);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(15,17,20,18)(16,22,25,26)(19,28,21,27)(23,24,30,29);(4,6)(15,18,20,17)(16,23,25,30)(19,28,21,27)(22,24,26,29);(7,10,11,9)(8,13,14,12)(15,19,20,21)(16,24,25,29)(17,27,18,28)(22,23,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352818);