Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8781 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #382 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
738 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);(3,4,6,8,5,7,9,10)(11,12)(13,21,24,17)(14,25,27,40)(15,20)(16,19)(18,26,31,41)(22,35)(23,34,38,30)(28,32)(29,36)(33,42)(37,39);(4,7)(8,10)(11,13,15,23,16,24,28,38)(12,17,20,30,19,21,32,34)(14,22,26,36,27,37,41,42)(18,29,25,39,31,33,40,35);(11,14)(12,18,19,31)(13,22)(15,26)(16,27)(17,29,21,33)(20,25,32,40)(23,36)(24,37)(28,41)(30,39,34,35)(38,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98331);