Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
530 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)(7,9)(10,11)(12,13)(14,15,19,24,34,43,32,41,20,25,18,23,17,22,33,42)(16,26,29,39,27,45,35,21,31,38,28,37,30,44,36,40);(6,7,8,9)(14,16)(15,21)(17,27)(18,35)(19,36)(20,31)(22,37)(23,44)(24,45)(25,40)(26,41)(28,32)(29,33)(30,34)(38,42)(39,43);(11,13)(14,17,20,34)(15,22,25,43)(16,27,31,30)(18,32,19,33)(21,37,40,39)(23,41,24,42)(26,45,38,44)(28,36,29,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5982);