Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11026 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #390 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,20,23,16)(13,24,27,17)(14,19)(15,18)(21,31)(22,36,28,32)(25,33)(26,37,29,34)(30,35)(38,39)(40,41);(7,9)(10,12)(11,16)(13,25)(14,28)(15,23)(17,33)(18,20)(19,36)(21,26)(22,30)(24,41)(27,40)(29,39)(31,34)(32,35)(37,38);(10,13)(11,17)(12,21)(14,29)(15,27)(16,31)(18,24)(19,37)(20,38)(22,40)(23,39)(25,28)(26,30)(32,41)(33,36)(34,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4014);