Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11297 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
578 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)(12,22,25,17,26,32,24,33)(13,27,28,18,30,34,29,35)(23,37,38,41,39,31,40,36);(6,7)(8,9)(10,12,14,25,16,26,15,24)(11,17,19,32,21,33,20,22)(13,23,29,38,30,39,28,40)(18,31,27,36,35,37,34,41);(7,9)(10,13,14,28,16,30,15,29)(11,18,19,34,21,35,20,27)(12,23,24,38,26,39,25,40)(17,31,22,36,33,37,32,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4285);