Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#22217 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(8,11)(9,13)(15,16,21,17,20,23,19,25)(18,22,27,24,28,30,29,26);(7,8)(9,12)(10,11)(13,14)(15,17,21,23,20,25,19,16)(18,26,27,22,28,24,29,30);(3,4,5,6)(8,11)(12,14)(15,18)(16,22)(17,24)(19,29)(20,28)(21,27)(23,30)(25,26);(7,9,10,13)(8,12,11,14)(15,19,20,21)(16,17,23,25)(18,27,28,29)(22,26,30,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283429);