Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1342 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)(11,12,17,21)(13,18,16,32)(14,24,27,41)(15,28,31,40)(19,34,36,26)(20,23,38,30)(22,39,25,37)(29,33,42,35);(3,4)(5,8)(6,7)(9,10)(11,13,17,16)(12,18,21,32)(14,25,27,22)(15,23,31,30)(19,35,36,33)(20,28,38,40)(24,37,41,39)(26,42,34,29);(4,7)(8,10)(11,14,17,27)(12,19,21,36)(13,22,16,25)(15,29,31,42)(18,33,32,35)(20,37,38,39)(23,34,30,26)(24,40,41,28);(3,5,6,9)(4,8,7,10)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530656);