Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11305 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #397 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,21)(13,18)(14,29)(15,20)(16,19)(17,23)(22,37)(24,38)(25,32)(26,31)(27,35)(28,36)(30,34)(33,39)(40,41);(10,12,16,25,15,24,14,23)(11,17,29,38,20,32,19,21)(13,26,27,39,28,40,30,22)(18,33,34,31,36,37,35,41);(10,13,14,27,15,28,16,30)(11,18,19,34,20,36,29,35)(12,22,23,26,24,39,25,40)(17,31,21,37,32,41,38,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4293);