Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33130 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2674 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);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16,21,26)(17,23,28,29)(18,22,27,30)(19,25,20,24);(7,9,10,13)(8,11,12,14)(15,17,20,27,21,28,19,18)(16,22,25,29,26,30,24,23);(8,12)(11,14);(4,6)(15,18,19,28,21,27,20,17)(16,23,24,30,26,29,25,22)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536916);