Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #99 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C22×C4⋊C4 | C22×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C5 ◁ C22×C5×C4⋊C4 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,14)(13,16)(15,17)(19,22);(11,14,16,15)(18,19,20,23,21,22,24,25)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 239);