Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×C5×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(5,6,9,14,10,13,8,12)(7,15,16,20,17,11,18,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13384);