Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33206 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,10,12)(9,13,14,11)(15,16)(17,26)(18,28)(19,23)(20,25)(21,22)(24,27)(29,30);(7,9,10,14)(8,11,12,13)(15,17,20,21)(16,22,25,26)(18,29,19,27)(23,30,28,24);(3,4,5,6)(15,18,20,19)(16,23,25,28)(17,27,21,29)(22,24,26,30);(15,19,20,18)(16,24,25,30)(17,27,21,29)(22,23,26,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537292);