Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ 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)(9,13)(10,12)(11,14)(15,16,18,20)(17,23,25,21)(19,22,26,28)(24,29,27,30);(3,4,6,5)(7,9,10,14)(8,11,12,13)(16,20)(17,24)(19,26)(21,29)(23,30)(25,27);(8,12)(11,13)(15,17,18,25)(16,21,20,23)(19,27,26,24)(22,29,28,30);(3,5,6,4)(8,12)(11,13)(16,20)(21,23)(22,28)(29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350720);