Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 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);(7,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,25)(20,24)(21,28)(22,26)(23,29)(27,30);(3,4,5,6)(7,9,11,10)(8,12,14,13)(15,17,19,16)(18,23,27,21)(20,24,26,22)(25,28,30,29);(4,6)(7,9,11,10)(8,13,14,12)(15,18,26,29,19,27,20,28)(16,21,24,30,17,23,22,25);(7,10,11,9)(8,13,14,12)(16,17)(21,23)(22,24)(25,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373721);