Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23601 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
994 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)(10,11,14,19,16,20,15,29)(12,21,23,32,25,38,24,17)(13,18,30,35,28,36,27,34)(22,37,40,31,39,33,26,41);(6,7)(8,9)(10,12,15,24,16,25,14,23)(11,17,19,21,20,32,29,38)(13,26,27,39,28,40,30,22)(18,33,35,41,36,37,34,31);(7,9)(10,13)(11,18)(12,22)(14,27)(15,30)(16,28)(17,31)(19,34)(20,36)(21,37)(23,26)(24,40)(25,39)(29,35)(32,41)(33,38);(10,14,16,15)(11,19,20,29)(12,23,25,24)(13,27,28,30)(17,21,32,38)(18,34,36,35)(22,26,39,40)(31,37,41,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16589);