Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #99 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| socle | C22×C5 | C2×#99 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11,14,19,15,20,12,17)(13,18,21,25,22,16,23,24);(6,7,8,9)(11,16,20,18)(17,24,19,25);(10,12,15,14)(11,17,20,19)(13,21,22,23)(16,24,18,25)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5573);