Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,17,29,28,39,27,19,16,22,14,20,15,21,31,38)(13,18,30,32,25,34,40,37,26,36,24,33,23,35,41,42);(3,5,7,6)(4,8,9,10)(11,13)(12,18)(14,24)(15,23)(16,26)(17,30)(19,37)(20,33)(21,35)(22,36)(25,28)(27,40)(29,32)(31,41)(34,39)(38,42);(3,6,7,5)(4,8,9,10)(12,19)(13,23)(14,27)(15,28)(17,31)(18,32)(20,21)(22,38)(24,30)(25,26)(29,39)(33,36)(34,42)(35,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406631);