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 | C2×#9671 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #390 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1160 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);(3,4,8,9,7,5,6,10)(11,12)(13,17,22,31)(14,24)(15,27)(16,19)(18,25)(20,28)(21,33,34,42)(23,37,36,30)(26,38)(29,40,41,35)(32,39);(3,5,8,10,7,4,6,9)(11,13,16,22)(12,17)(14,21,26,34)(15,23,28,36)(18,29)(19,31)(20,30)(24,33)(25,35,39,40)(27,37)(32,41)(38,42);(3,6,7,8)(4,9,5,10)(11,14,15,25,16,26,28,39)(12,18,20,24,19,32,27,38)(13,21,23,35,22,34,36,40)(17,29,30,33,31,41,37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111629);