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×#11368 |
| 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)(6,7)(8,9)(10,11,16,19,15,20,14,29)(12,21,23,17,24,33,30,32)(13,18,26,36,27,37,28,35)(22,34,25,31,40,38,39,41);(7,9)(10,12,16,23,15,24,14,30)(11,17,19,33,20,32,29,21)(13,25,28,22,27,39,26,40)(18,34,35,41,37,38,36,31);(7,9)(10,13,16,28,15,27,14,26)(11,18,19,35,20,37,29,36)(12,22,23,39,24,40,30,25)(17,31,33,34,32,41,21,38)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4356);