Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33476 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1376 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);(3,4)(5,10)(6,8)(7,9)(11,12,14,18,15,19,24,30,16,20,25,31,26,32,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42);(3,5,7,6)(4,8,9,10);(3,6,7,5)(4,8,9,10)(11,13,14,22,15,23,24,35,16,21,25,33,26,34,36,41)(12,17,18,28,19,29,30,39,20,27,31,37,32,38,40,42);(13,21)(17,27)(22,33)(23,34)(28,37)(29,38)(35,41)(39,42)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529838);