Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#21756 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,20,16,19,28,33)(13,17,23,31,22,30,38,41)(14,24,25,34,26,18,27,32)(21,35,37,42,36,29,39,40);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,29)(19,30)(20,31)(24,35)(25,37)(26,36)(27,39)(28,38)(32,40)(33,41)(34,42);(5,10)(7,9)(11,14,16,26)(12,18,19,24)(13,21,22,36)(15,27,28,25)(17,29,30,35)(20,34,33,32)(23,39,38,37)(31,42,41,40);(4,8)(7,9)(13,22)(17,30)(21,36)(23,38)(29,35)(31,41)(37,39)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281334);