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×C4 |
| direct product | C5×#21250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1434 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)(7,9)(10,11,15,20,16,19,14,28)(12,17,23,32,22,33,29,31)(13,24,27,34,25,38,26,18)(21,35,37,30,36,41,39,40);(6,7,8,9)(10,12,14,23,16,22,15,29)(11,17,20,31,19,33,28,32)(13,21,26,37,25,36,27,39)(18,30,24,35,34,40,38,41);(10,13,14,26,16,25,15,27)(11,18,20,24,19,34,28,38)(12,21,23,37,22,36,29,39)(17,30,31,35,33,40,32,41);(13,25)(18,34)(21,36)(24,38)(26,27)(30,40)(35,41)(37,39)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14238);