Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25 |
| direct product | C22×C5×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C42 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | #97 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(21,22,23,24,25);(3,4);(5,6,7,11,8,13,15,19,9,12,14,20,16,10,17,18);(6,10,12,13)(7,14)(11,19,20,18)(15,17)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26315);