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×#11295 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #400 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
574 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)(10,11,15,20,16,21,14,19)(12,22,25,17,26,33,24,34)(13,27,29,18,31,36,30,37)(23,39,40,35,41,32,28,38);(6,7)(8,9)(10,12,14,25,16,26,15,24)(11,17,19,33,21,34,20,22)(13,28,30,23,31,40,29,41)(18,35,27,32,37,38,36,39);(7,9)(10,13,14,29,16,31,15,30)(11,18,19,36,21,37,20,27)(12,23,24,40,26,41,25,28)(17,32,22,38,34,39,33,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4283);