Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10871 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #390 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
746 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)(12,21)(13,26)(14,30)(15,19)(16,20)(17,23)(18,27)(22,31)(24,33)(25,32)(28,34)(29,40)(35,37)(36,39)(38,41);(10,12,15,23,16,25,14,24)(11,17,19,32,20,33,30,21)(13,22,28,36,29,38,27,37)(18,31,26,39,34,41,40,35);(10,13,14,27,16,29,15,28)(11,18,19,26,20,34,30,40)(12,22,24,37,25,38,23,36)(17,31,32,39,33,41,21,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3859);