Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1931 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#121 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4);(5,6,8,7)(10,14)(11,18)(12,15)(16,23)(17,21)(20,24);(5,7,8,6)(9,10,15,21,13,17,12,14)(11,16,22,23,18,24,19,20);(9,11,12,19,13,18,15,22)(10,16,14,20,17,24,21,23)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261553);