Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
874 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)(10,11,16,21)(12,22,24,33)(13,27,29,35)(14,20,15,19)(17,26,32,25)(18,30,34,28)(23,37,38,31)(36,40,41,39);(10,12,14,25,16,24,15,26)(11,17,19,22,21,32,20,33)(13,23,28,39,29,38,30,40)(18,31,27,36,34,37,35,41);(10,13)(11,18)(12,23)(14,28)(15,30)(16,29)(17,31)(19,27)(20,35)(21,34)(22,36)(24,38)(25,39)(26,40)(32,37)(33,41);(6,8,9,7)(12,24)(17,32)(22,33)(23,38)(25,26)(31,37)(36,41)(39,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14544);