Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21476 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1322 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)(6,7,9,8)(10,11,15,19)(12,20,23,34)(13,17,24,32)(14,26,29,18)(16,27,31,22)(21,37,38,36)(25,35,28,33)(30,40,41,39);(10,12)(11,16)(13,21)(14,27)(15,23)(17,30)(18,34)(19,31)(20,26)(22,29)(24,38)(25,40)(28,39)(32,41)(33,36)(35,37);(10,13,15,24)(11,17,19,32)(12,21,23,38)(14,28,29,25)(16,30,31,41)(18,35,26,33)(20,36,34,37)(22,40,27,39);(6,8,9,7)(13,24)(17,32)(21,38)(25,28)(30,41)(33,35)(36,37)(39,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14464);