Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,8,7,5,10,9)(11,12)(13,22)(14,26,29,36)(15,30)(16,20)(17,21)(18,24)(19,28,35,27)(23,38,40,32)(25,33)(31,34)(37,39,42,41);(3,5)(4,7)(6,8)(9,10)(11,13,15,24,17,25,16,31)(12,18,20,22,21,34,30,33)(14,23,28,41,29,40,27,39)(19,32,36,37,35,38,26,42);(3,4)(5,7)(6,8)(9,10)(11,14)(12,19)(13,23)(15,28)(16,27)(17,29)(18,32)(20,36)(21,35)(22,37)(24,41)(25,40)(26,30)(31,39)(33,42)(34,38)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106835);