Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11345 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
698 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)(7,9)(10,11,14,19,15,18,26,32)(12,16,22,28,21,29,34,39)(13,17,25,31,24,30,37,40)(20,27,33,38,23,35,36,41);(10,12)(11,16)(13,23,24,20)(14,22)(15,21)(17,27,30,35)(18,29)(19,28)(25,36,37,33)(26,34)(31,38,40,41)(32,39);(6,7,8,9)(10,13)(11,17)(12,20)(14,25)(15,24)(16,27)(18,30)(19,31)(21,23)(22,33)(26,37)(28,38)(29,35)(32,40)(34,36)(39,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4333);