Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13529 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 46 points:
(1,2);(7,8,10,12)(9,13,14,11)(15,16,18,23,32,40,43,36,20,25,33,35,19,24,30,21)(17,26,27,41,34,46,45,38,29,42,44,37,28,39,31,22);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(16,21)(18,30)(19,32)(22,26)(23,24)(25,36)(27,31)(28,34)(33,43)(35,40)(37,46)(38,42)(39,41)(44,45);(4,6)(15,17,20,29)(16,22,25,38)(18,31,33,45)(19,34,32,28)(21,26,36,42)(23,39,35,46)(24,41,40,37)(27,43,44,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406720);