Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8265 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C23 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(10,11,14,16,13,15,17,12)(18,19,21,20,22,23,25,24);(7,9)(11,15)(12,16)(18,19,21,20,22,23,25,24);(10,12,17,15,13,16,14,11)(18,20,21,23,22,24,25,19)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1253);