Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,14,17,15,16,13,18)(19,20)(21,27,29,24)(22,30)(23,26)(25,31)(28,32,34,33);(4,7)(5,10)(8,9)(12,16)(13,14)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(11,13,15,14)(12,17,16,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68878);