Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21550 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,19,16,20,14,30)(12,21,25,31,22,36,24,17)(13,18,28,34,29,35,27,41)(23,37,39,33,26,40,38,32);(10,12,14,24,16,22,15,25)(11,17,19,21,20,31,30,36)(13,26,27,39,29,23,28,38)(18,33,34,40,35,32,41,37);(7,9)(12,22)(17,31)(21,36)(23,26)(24,25)(32,33)(37,40)(38,39);(10,13,15,28,16,29,14,27)(11,18,19,34,20,35,30,41)(12,23,25,39,22,26,24,38)(17,32,21,37,31,33,36,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14538);