Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33766 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,19,24)(16,20,25,33)(17,21,30,37)(18,31,29,23)(22,38,36,28)(26,32,39,43)(27,42,40,35)(34,45,44,41);(6,8)(7,10)(9,13)(11,12)(14,16,19,25)(15,20,24,33)(17,26,30,39)(18,27,29,40)(21,32,37,43)(22,34,36,44)(23,35,31,42)(28,41,38,45);(8,13)(10,12)(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(26,41)(27,40)(30,38)(31,36)(33,43)(34,35)(39,45)(42,44);(7,11)(10,12)(16,25)(20,33)(26,39)(27,40)(32,43)(34,44)(35,42)(41,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26754);