Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2047 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#121 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4);(5,6)(7,8)(9,10,14,18)(11,19,22,34)(12,23,26,16)(13,27,29,17)(15,28,31,21)(20,36,24,35)(25,33,40,38)(30,37,32,39);(9,11,14,22)(10,15,18,31)(12,24,26,20)(13,28,29,21)(16,32,23,30)(17,34,27,19)(25,37,40,39)(33,35,38,36);(6,8)(9,12,13,25,14,26,29,40)(10,16,17,33,18,23,27,38)(11,20,21,37,22,24,28,39)(15,30,19,35,31,32,34,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266292);