Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#22465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1312 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)(10,11,15,21,17,13,16,20)(12,22,26,23,27,34,25,18)(14,19,31,36,32,28,30,37)(24,38,29,39,41,35,40,33);(10,12)(11,18)(13,23)(14,29)(15,26)(16,25)(17,27)(19,35)(20,34)(21,22)(24,30)(28,38)(31,41)(32,40)(33,36)(37,39);(7,9)(10,13,17,11)(12,23,27,18)(14,28,32,19)(15,21,16,20)(22,25,34,26)(24,39,41,33)(29,38,40,35)(30,37,31,36);(10,14)(11,19)(12,24)(13,28)(15,31)(16,30)(17,32)(18,33)(20,37)(21,36)(22,38)(23,39)(25,40)(26,29)(27,41)(34,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15453);