Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8157 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1318 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);(11,12,15,19,16,20,27,33)(13,21,24,35,23,17,28,30)(14,18,25,31,26,32,38,41)(22,34,37,42,36,29,39,40);(3,4,5,8,6,7,9,10)(11,13)(12,17,20,21)(14,22)(15,24)(16,23)(18,29,32,34)(19,30,33,35)(25,37)(26,36)(27,28)(31,40,41,42)(38,39);(4,7)(8,10)(11,14,15,25,16,26,27,38)(12,18,19,31,20,32,33,41)(13,22,24,37,23,36,28,39)(17,29,30,40,21,34,35,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89238);