Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16260 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#16399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4982 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)(12,14)(15,16,18,20)(17,19,21,22);(7,8,10,9)(17,21)(19,22);(8,9)(11,12,13,14)(15,17)(16,19)(18,21)(20,22);(3,4)(5,6)(7,9,10,8)(15,16)(17,19)(18,20)(21,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518104);