Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6)(7,8,12,16)(9,18,14,21)(10,19,20,13)(11,17,22,15);(7,9)(8,10)(11,19)(12,14)(13,22)(15,21)(16,20)(17,18);(4,6)(7,10,12,20)(8,13,16,19)(9,15,14,17)(11,18,22,21);(7,10,12,20)(8,14,16,9)(11,21,22,18)(13,17,19,15)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345091);