Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9592 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #388 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,8)(5,10)(6,9)(11,12,14,16,15,19,23,18)(13,20,22,26,21,17,25,24);(3,4,9,10,7,5,6,8)(12,16,19,18)(13,21)(17,26,20,24)(22,25);(3,5,6,10,7,4,9,8)(11,13,14,22,15,21,23,25)(12,17,18,26,19,20,16,24)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110316);