Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22899 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2358 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7,11,16,13,17,12,15)(8,18,20,21,10,9,19,14);(6,8,11,20,13,10,12,19)(7,14,16,18,17,21,15,9);(6,9)(7,8)(10,17)(11,14)(12,21)(13,18)(15,19)(16,20);(6,10)(7,9)(8,13)(11,19)(12,20)(14,16)(15,21)(17,18)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15887);