Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25558 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3564 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);(4,6)(7,8)(9,13)(10,12)(11,14)(15,16,21,25)(17,22,26,23)(18,27,28,30)(19,24,20,29);(7,9,10,14)(8,11,12,13)(15,17)(16,22)(18,19)(20,28)(21,26)(23,25)(24,27)(29,30);(3,4,5,6)(15,18,19,17,21,28,20,26)(16,23,29,30,25,22,24,27);(8,12)(11,13)(15,19,21,20)(16,24,25,29)(17,18,26,28)(22,27,23,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359338);