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×#25711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3256 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)(7,8,10,11)(9,13,14,12)(15,16,20,25)(17,23,28,22)(18,29,27,30)(19,26,21,24);(8,11)(12,13)(15,17,20,28)(16,22,25,23)(18,19,27,21)(24,30,26,29);(3,4,5,6)(15,18,21,17,20,27,19,28)(16,23,26,29,25,22,24,30);(7,9,10,14)(8,12,11,13)(15,19,20,21)(16,24,25,26)(17,27,28,18)(22,30,23,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357322);