Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,9)(14,15,19,24)(16,25,28,43)(17,21,32,39)(18,33,31,23)(20,35,37,26)(22,40,38,30)(27,44,34,41)(29,42,45,36);(7,9)(10,11)(12,13)(14,16,19,28)(15,20,24,37)(17,29,32,45)(18,34,31,27)(21,25,39,43)(22,41,38,44)(23,42,33,36)(26,40,35,30);(11,13)(15,21)(17,30)(18,31)(20,25)(22,23)(24,39)(26,29)(27,34)(32,40)(33,38)(35,45)(36,44)(37,43)(41,42)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6317);