Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1372 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);(3,4,5,7,6,9,10,8)(11,12)(13,22,25,33)(14,26)(15,20)(16,30)(17,21)(18,24,34,29)(19,27)(23,38,41,32)(28,35)(31,36)(37,39,42,40);(4,7,9,8)(11,13,15,29,17,25,16,24)(12,18,20,33,21,34,30,22)(14,23,27,40,28,41,31,39)(19,32,35,42,36,38,26,37);(11,14)(12,19)(13,23)(15,27)(16,31)(17,28)(18,32)(20,35)(21,36)(22,37)(24,39)(25,41)(26,30)(29,40)(33,42)(34,38)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106457);