Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8180 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle, prime radical | C23×C5 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C43×C5 ◁ G | |
530 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)(13,16)(15,17)(18,19)(20,21)(22,25)(23,24);(11,14,16,15)(18,20,22,24,23,25,21,19);(6,8,9,7)(19,24)(20,25)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1168);