Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×C5×#382 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | C23 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| socle, prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ G | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(6,7)(9,10,11,12)(15,19)(17,20);(5,6,8,7)(13,14,16,18)(15,17,19,20);(5,7,8,6)(13,15)(14,17)(16,19)(18,20)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6761);