Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,11,15,12,9,16,14)(10,18,20,17,21,22,19,13);(3,4)(5,6)(7,9)(8,12)(10,17)(11,14)(13,21)(15,16)(18,19)(20,22);(4,6)(7,10,12,21)(8,13,9,17)(11,19,16,20)(14,18,15,22);(7,9)(8,12)(10,17)(11,14)(13,21)(15,16)(18,19)(20,22)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345119);