Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,7,9)(5,8,6,10)(11,12)(13,24)(14,19)(15,32)(16,22)(17,23)(18,27)(20,31)(21,34)(25,38)(26,35)(28,36)(29,42)(30,39)(33,37)(40,41);(3,5,7,6)(4,8,9,10)(11,13)(12,18)(14,29)(15,26)(16,33)(17,28)(19,38)(20,35)(21,36)(22,24)(23,37)(25,31)(27,34)(30,41)(32,40)(39,42);(3,6,7,5)(4,8,9,10)(11,14,17,15)(12,19,23,32)(13,25,28,41)(16,30,34,31)(18,35,37,42)(20,22,39,21)(24,38,36,40)(26,33,29,27);(11,15,17,14)(12,20,23,39)(13,26,28,29)(16,30,34,31)(18,35,37,42)(19,22,32,21)(24,40,36,38)(25,33,41,27)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346441);