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×#33676 |
| 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 | |
1560 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)(5,8)(6,7)(9,10)(11,12,17,22)(13,18,26,34)(14,20,16,30)(15,19,29,36)(21,27,35,28)(23,32,25,40)(24,31,39,42)(33,37,41,38);(4,7)(8,10)(11,13,17,26)(12,18,22,34)(14,23,16,25)(15,24,29,39)(19,31,36,42)(20,32,30,40)(21,33,35,41)(27,37,28,38);(12,19)(14,16)(15,28)(18,31)(20,21)(22,36)(23,25)(24,38)(27,29)(30,35)(32,33)(34,42)(37,39)(40,41);(3,5,6,9)(4,8,7,10)(11,14,17,16)(12,20,22,30)(13,23,26,25)(15,27,29,28)(18,32,34,40)(19,35,36,21)(24,37,39,38)(31,41,42,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530273);