Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25847 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,11,14)(9,13,10,12)(15,16)(17,23)(18,29)(19,26)(20,25)(21,24)(22,27)(28,30);(7,9,11,10)(8,12,14,13)(15,17)(16,22)(18,19)(20,27)(21,28)(23,25)(24,29)(26,30);(3,4,5,6)(15,18,19,17,20,28,21,27)(16,23,24,29,25,22,26,30);(7,10,11,9)(8,13,14,12)(15,19,20,21)(16,24,25,26)(17,18,27,28)(22,29,23,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352933);