Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21548 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
914 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)(6,7)(8,9)(10,11,15,19,16,20,14,29)(12,21,25,30,22,34,24,17)(13,18,27,32,28,33,26,39)(23,35,38,40,36,41,37,31);(10,12,14,24,16,22,15,25)(11,17,19,21,20,30,29,34)(13,23,26,37,28,36,27,38)(18,31,32,35,33,40,39,41);(7,9)(12,22)(17,30)(21,34)(23,36)(24,25)(31,40)(35,41)(37,38);(10,13,15,27,16,28,14,26)(11,18,19,32,20,33,29,39)(12,23,25,38,22,36,24,37)(17,31,21,35,30,40,34,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14536);