Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11272 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #389 |
| 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)(6,7,9,8)(10,11,15,19,16,21,14,20)(12,22,25,33,26,36,24,17)(13,27,30,35,31,37,29,18)(23,39,40,34,41,32,28,38);(6,8,9,7)(10,12,14,24,16,26,15,25)(11,17,19,22,21,33,20,36)(13,28,29,41,31,40,30,23)(18,34,27,32,35,38,37,39);(7,8)(10,13,14,29,16,31,15,30)(11,18,19,27,21,35,20,37)(12,23,24,28,26,41,25,40)(17,32,22,38,33,39,36,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4260);