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×#24614 |
| 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 | |
2172 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)(9,14)(10,12)(11,13)(15,16,19,23,18,22,27,21)(17,24,26,29,25,20,28,30);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,18,25)(16,20,22,24)(19,26,27,28)(21,29,23,30);(4,6)(7,10,11,9)(8,12,13,14)(16,21)(19,27)(20,29)(22,23)(24,30)(26,28);(8,13)(12,14)(16,22)(17,25)(21,23)(26,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360335);