Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26839 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 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)(15,16,20,24)(17,22,27,30)(18,28,26,21)(19,29,25,23);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17,25,18,20,27,19,26)(16,21,29,30,24,28,23,22);(4,6)(7,10,11,9)(8,12,13,14)(15,17,19,26,20,27,25,18)(16,21,23,22,24,28,29,30);(8,13)(12,14)(15,18,25,27,20,26,19,17)(16,22,29,21,24,30,23,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351066);