Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#16549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(8,9)(15,16)(17,21)(18,20)(19,22);(11,12,13,14)(15,16,18,20)(17,19,22,21);(7,8,10,9)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22);(3,4)(5,6)(7,9,10,8)(15,17)(16,19)(18,22)(20,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518300);