Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21072 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
1018 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)(7,8)(10,11,15,20,16,19,14,28)(12,17,24,31,22,32,23,38)(13,25,26,18,27,34,29,33)(21,35,37,30,36,40,39,41);(6,7,9,8)(10,12,15,24,16,22,14,23)(11,17,20,31,19,32,28,38)(13,21,26,37,27,36,29,39)(18,30,34,40,33,41,25,35);(6,8,9,7)(10,13)(11,18)(12,21)(14,29)(15,26)(16,27)(17,30)(19,33)(20,34)(22,36)(23,39)(24,37)(25,28)(31,40)(32,41)(35,38);(11,19)(12,22)(18,33)(20,28)(21,36)(23,24)(25,34)(37,39)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14060);