Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33814 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1374 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);(4,7)(5,9)(11,12,16,21)(13,17,24,32)(14,18,27,34)(15,28,26,20)(19,35,33,25)(22,29,38,41)(23,39,37,31)(30,42,40,36);(3,4)(5,8)(6,7)(9,10)(11,13,16,24)(12,17,21,32)(14,22,27,38)(15,23,26,37)(18,29,34,41)(19,30,33,40)(20,31,28,39)(25,36,35,42);(4,7)(8,10)(12,18)(14,25)(15,26)(17,29)(19,20)(21,34)(22,36)(23,37)(27,35)(28,33)(30,31)(32,41)(38,42)(39,40);(3,5,6,9)(4,8,7,10)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530648);