Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C8 |
| direct product | C5×#8793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #382 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,8)(9,10)(12,13)(14,15)(16,24,28,20)(17,29,31,44)(18,23)(19,22)(21,30,35,39)(25,32)(26,40)(27,37,38,34)(33,41)(36,45)(42,43);(6,7,9,8,11,13,10,12)(14,16,25,38,19,28,18,27)(15,20,32,37,22,24,23,34)(17,26,39,45,31,42,30,41)(21,33,44,40,35,36,29,43);(6,8,10,7,11,12,9,13)(14,17,19,31)(15,21)(16,26,28,42)(18,30,25,39)(20,33)(22,35)(23,29)(24,36)(27,41,38,45)(32,44)(34,43)(37,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1781);