Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24072 |
| 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 | |
3226 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)(15,16,21,24)(17,22,26,30)(18,27,25,29)(19,23,20,28);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(15,17)(16,22)(18,19)(20,25)(21,26)(23,27)(24,30)(28,29);(4,6)(7,10,11,9)(8,12,13,14)(15,18)(16,22)(17,19)(20,26)(21,25)(23,29)(24,30)(27,28);(8,13)(12,14)(15,19,21,20)(16,23,24,28)(17,25,26,18)(22,29,30,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360088);