Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#22412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#550 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,14,19,15,18,12,16)(13,21,20,25,22,17,23,24);(6,7)(8,9)(10,12,15,14)(11,16,18,19)(13,20,22,23)(17,24,21,25);(10,13,15,22)(11,17,18,21)(12,20,14,23)(16,24,19,25);(7,9)(10,12,15,14)(11,16,18,19)(13,20,22,23)(17,24,21,25)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15400);