Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#127 | C22 |
| 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);(7,8,10,11)(9,13,14,12)(16,21)(17,24)(18,20)(22,29)(25,27)(28,30);(3,4,5,6)(8,11)(12,13)(15,16)(17,22)(18,26)(19,21)(20,23)(24,30)(25,28)(27,29);(4,6)(15,17,20,24,19,25,18,27)(16,22,23,30,21,28,26,29);(7,9,10,14)(8,12,11,13)(15,18,19,20)(16,23,21,26)(17,24,25,27)(22,29,28,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371200);