Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33409 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C22×D8 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
674 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)(12,21)(13,18)(14,28)(15,30)(16,20)(17,25)(19,22)(23,34)(24,35)(26,40)(27,33)(29,41)(31,38)(32,39)(36,37);(10,12,22,37,15,26,29,38,16,23,27,39,13,24,14,25)(11,17,28,35,18,32,33,34,20,31,41,40,30,36,19,21);(12,23)(17,31)(21,34)(24,26)(25,38)(32,36)(35,40)(37,39);(10,13,16,15)(11,18,20,30)(12,24,23,26)(14,27,29,22)(17,32,31,36)(19,28,33,41)(21,35,34,40)(25,39,38,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26397);