Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
890 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)(12,17,23,31,24,32,22,38)(13,25,28,33,26,39,27,18)(21,34,37,40,35,41,36,30);(7,9)(10,12,15,23,16,24,14,22)(11,17,19,31,20,32,29,38)(13,21,28,37,26,35,27,36)(18,30,25,34,33,40,39,41);(10,13,14,27,16,26,15,28)(11,18,19,25,20,33,29,39)(12,21,22,36,24,35,23,37)(17,30,31,34,32,40,38,41);(13,26)(18,33)(21,35)(25,39)(27,28)(30,40)(34,41)(36,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14152);