Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C8 |
| direct product | C2×#9839 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4,5,6)(8,11)(9,12)(13,14)(15,16)(17,23)(18,20)(19,27)(21,25)(22,26)(24,29)(28,30);(4,6)(7,8,12,14,10,13,9,11)(16,20)(17,24)(22,27)(25,30);(15,17,19,25,18,24,26,30)(16,21,22,23,20,28,27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113359);