Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33726 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C22×D8 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,22)(14,19)(15,29)(16,31)(17,21)(18,25)(20,23)(24,35)(26,41)(27,37)(28,34)(30,42)(32,39)(33,40)(36,38);(3,5,6,10)(4,7,8,9);(11,13,23,38,16,26,30,40,17,27,28,39,14,24,15,25)(12,18,29,35,19,32,34,37,21,33,42,41,31,36,20,22);(11,14,17,16)(12,19,21,31)(13,24,27,26)(15,28,30,23)(18,32,33,36)(20,29,34,42)(22,35,37,41)(25,39,40,38)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530406);