Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4454 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)(9,13)(10,12)(11,14)(15,16)(17,21)(18,23)(19,28)(20,24)(22,25)(26,30)(27,29);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,17)(16,21)(18,27)(19,25)(20,26)(22,28)(23,29)(24,30);(4,6)(15,17)(16,22)(18,25)(19,27)(20,26)(21,28)(23,30)(24,29);(8,12)(11,13)(15,18,20,19)(16,23,24,28)(17,25,26,27)(21,22,30,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358879);