Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,20)(13,19,23,17)(14,25,27,41)(16,30,29,21)(18,33,35,28)(22,37,26,40)(24,32,39,36)(31,38,34,42);(3,5)(4,7)(6,10)(8,9)(11,13,15,23)(12,17,20,19)(14,26,27,22)(16,24,29,39)(18,34,35,31)(21,32,30,36)(25,37,41,40)(28,42,33,38);(11,14,16,28,15,27,29,33)(12,18,21,25,20,35,30,41)(13,22,24,38,23,26,39,42)(17,31,32,40,19,34,36,37);(5,10)(7,9)(11,13)(12,19)(14,26)(15,23)(16,24)(17,20)(18,31)(21,36)(22,27)(25,40)(28,42)(29,39)(30,32)(33,38)(34,35)(37,41);(4,8)(7,9)(12,20)(13,23)(18,35)(21,30)(22,26)(24,39)(25,41)(38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484830);