Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33118 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2738 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);(4,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,21)(19,23)(22,26)(24,27)(25,28)(29,30);(3,4,6,5)(7,9,10,13)(8,11,12,14)(17,24)(20,27)(25,30)(28,29);(8,12)(11,14)(15,17,19,24)(16,20,23,27)(18,25,26,30)(21,28,22,29);(3,5,6,4)(16,21)(18,26)(20,28)(22,23)(25,30)(27,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536862);