Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#8793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #382 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,5)(6,7)(9,10)(11,12)(13,21,25,17)(14,26,28,41)(15,20)(16,19)(18,27,32,36)(22,29)(23,37)(24,34,35,31)(30,38)(33,42)(39,40);(3,4,6,5,8,10,7,9)(11,13,22,35,16,25,15,24)(12,17,29,34,19,21,20,31)(14,23,36,42,28,39,27,38)(18,30,41,37,32,33,26,40);(3,5,7,4,8,9,6,10)(11,14,16,28)(12,18)(13,23,25,39)(15,27,22,36)(17,30)(19,32)(20,26)(21,33)(24,38,35,42)(29,41)(31,40)(34,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98418);