Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11999 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #106 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#106 |
| prime radical | C23×C5 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
234 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)(6,7,8,10,9,11,12,13)(14,15)(16,25,28,39)(17,22)(18,31)(19,34)(20,24)(21,27,36,41)(23,32)(26,38,29,43)(30,37)(33,44)(35,40,45,42);(14,16,18,27,19,29,32,42,20,28,33,41,17,26,30,40)(15,21,23,25,22,35,31,38,24,36,37,39,34,45,44,43);(14,17,20,19)(15,22,24,34)(16,26,28,29)(18,30,33,32)(21,35,36,45)(23,31,37,44)(25,38,39,43)(27,40,41,42)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4987);