Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C5×AΣL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| socle | C22×C5 | C2×#99 |
| prime radical | C2×C5×#120 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×D4 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4)(13,14,15,16,17);(5,6)(7,11,12,9)(8,10);(6,9)(10,11)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5607);