Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#35556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(4,5)(7,8,11,9)(10,12,14,13);(3,4,6,5)(7,8,11,9)(10,13,14,12)(15,16)(17,18)(19,20)(21,22);(3,5,6,4)(8,9)(12,13)(16,20)(17,21);(7,9,11,8)(10,13,14,12)(15,17,19,21)(16,18,20,22);(7,10)(8,12)(9,13)(11,14)(15,18)(16,21)(17,20)(19,22)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536133);