Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1458 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)(7,9)(10,11,15,21,16,20,14,19)(12,17,23,31,24,33,29,32)(13,25,27,35,28,34,26,18)(22,36,37,30,38,40,39,41);(6,7,8,9)(10,12,15,29,16,24,14,23)(11,17,19,31,20,33,21,32)(13,22,27,39,28,38,26,37)(18,30,34,40,35,41,25,36);(10,13,14,26,16,28,15,27)(11,18,21,25,20,35,19,34)(12,22,23,37,24,38,29,39)(17,30,32,36,33,41,31,40);(10,14,16,15)(11,19,20,21)(12,23,24,29)(13,26,28,27)(17,31,33,32)(18,34,35,25)(22,37,38,39)(30,40,41,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14236);