Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9645 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(4,6)(7,8,12,13,11,9,10,14)(15,16)(17,23)(18,26)(19,21)(20,24)(22,27)(25,29)(28,30);(7,9,12,14,11,8,10,13)(15,17,18,24,19,25,27,30)(16,20,22,23,21,28,26,29);(3,4,5,6)(7,10,11,12)(8,13,9,14)(16,21)(20,28)(22,26)(23,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111241);