Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1989 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4);(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,19,23,20,22,24,21);(5,6,7,8)(10,13)(14,15)(18,21)(19,24)(22,23);(9,11,12,16)(10,14,13,15)(18,22)(21,23)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261941);