Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#9562 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
398 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)(6,7,9,8)(10,11,12,14)(13,15,17,16)(19,24)(20,25)(21,22);(6,8,9,7)(11,14)(15,16)(18,19,21,24,23,25,22,20);(11,15)(13,17)(14,16)(18,20,22,25,23,24,21,19)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2550);