Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33630 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1546 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)(11,12,15,18)(13,23,25,40)(14,22,17,20)(16,29,28,37)(19,35,33,26)(21,27,34,31)(24,41,32,38)(30,39,42,36);(3,5,6,10)(4,7,8,9)(12,18)(19,33)(20,22)(21,34)(23,40)(29,37)(36,39)(38,41);(11,13)(12,19)(14,24)(15,25)(16,30)(17,32)(18,33)(20,36)(21,38)(22,39)(23,29)(26,31)(27,35)(28,42)(34,41)(37,40);(11,14,15,17)(12,20,18,22)(13,24,25,32)(16,27,28,31)(19,36,33,39)(21,37,34,29)(23,38,40,41)(26,30,35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530201);