Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,16,22)(12,24,13,19)(14,23,17,20)(15,30,32,38)(18,28,35,26)(21,29,39,33)(25,36,27,37)(31,40,41,34);(10,12,16,13)(11,18,22,35)(14,25,17,27)(15,31,32,41)(19,30,24,38)(20,34,23,40)(21,37,39,36)(26,29,28,33);(6,8,9,7)(10,13,16,12)(11,19,22,24)(14,25,17,27)(15,28,32,26)(18,30,35,38)(20,37,23,36)(21,34,39,40)(29,31,33,41);(10,14,16,17)(11,20,22,23)(12,25,13,27)(15,29,32,33)(18,34,35,40)(19,36,24,37)(21,38,39,30)(26,31,28,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26448);