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) = C24×C4 |
| direct product | C5×#21100 |
| 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 | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
1030 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)(12,21,25,32,23,35,24,17)(13,26,29,33,27,40,28,18)(22,36,38,41,39,31,37,34);(10,12)(11,17,20,32)(13,22)(14,24)(15,25)(16,23)(18,31,33,36)(19,21,30,35)(26,34,40,41)(27,39)(28,37)(29,38);(7,9)(10,13,14,28,16,27,15,29)(11,18,19,26,20,33,30,40)(12,22,24,37,23,39,25,38)(17,31,21,34,32,36,35,41);(12,23)(13,27)(17,32)(18,33)(21,35)(24,25)(26,40)(28,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14088);