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×#33649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,22)(13,23,26,40)(14,21,16,19)(15,28,31,36)(18,33,35,25)(20,27,38,30)(24,41,32,37)(29,39,42,34);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(11,13)(12,18)(14,24)(15,29)(16,32)(17,26)(19,34)(20,37)(21,39)(22,35)(23,28)(25,30)(27,33)(31,42)(36,40)(38,41);(11,14,17,16)(12,19,22,21)(13,24,26,32)(15,27,31,30)(18,34,35,39)(20,36,38,28)(23,37,40,41)(25,29,33,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530235);