Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#20393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5470 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);(3,4,6,5)(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,20)(19,22);(7,9)(8,12,14,15)(10,16)(11,17)(13,19,20,21)(18,22);(3,5,6,4)(10,18)(13,20)(16,22)(19,21);(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277012);