Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6348 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);(4,7)(8,9)(11,12,16,20)(13,22,24,18)(14,26,28,36)(15,21,17,30)(19,31,35,27)(23,38,39,42)(25,34,29,33)(32,41,37,40);(3,4,6,7)(5,9,10,8)(11,13,16,24)(12,18,20,22)(14,23,28,39)(15,29,17,25)(19,32,35,37)(21,34,30,33)(26,42,36,38)(27,41,31,40);(11,14)(12,19)(13,23)(15,27)(16,28)(17,31)(18,32)(20,35)(21,36)(22,37)(24,39)(25,40)(26,30)(29,41)(33,42)(34,38);(3,5,6,10)(4,8,7,9)(12,20)(13,24)(19,35)(21,30)(23,39)(25,29)(26,36)(40,41);(11,15,16,17)(12,21,20,30)(13,25,24,29)(14,27,28,31)(18,33,22,34)(19,36,35,26)(23,40,39,41)(32,42,37,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483686);