Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,6)(11,12,16,21)(13,22,25,40)(14,18,29,36)(15,30,28,20)(17,32,34,23)(19,37,35,27)(24,41,31,38)(26,39,42,33);(4,6)(7,8)(9,10)(11,13,16,25)(12,17,21,34)(14,26,29,42)(15,31,28,24)(18,22,36,40)(19,38,35,41)(20,39,30,33)(23,37,32,27);(8,10)(12,18)(14,27)(15,28)(17,22)(19,20)(21,36)(23,26)(24,31)(29,37)(30,35)(32,42)(33,41)(34,40)(38,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406037);