Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
322 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)(6,7,9,12,10,8,13,11)(14,15)(16,25,28,41)(17,21)(18,32)(19,24)(20,27,36,45)(22,30)(23,31)(26,39,29,42)(33,35,43,37)(34,38)(40,44);(6,8)(7,10)(9,11)(12,13)(14,16)(15,20)(17,29)(18,33)(19,28)(21,37)(22,39)(23,35)(24,36)(25,38)(26,31)(27,44)(30,45)(32,41)(34,43)(40,42);(7,8)(11,12)(14,17,19,31)(15,21,24,23)(16,26,28,29)(18,30,34,44)(20,35,36,37)(22,38,40,32)(25,39,41,42)(27,43,45,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4670);