Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#7182 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C8 | C4×C5×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| socle | C22×C5 | D4⋊C8 |
| prime radical | C23×C5 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,13,19,15,20,23,17,16,21,24,18,14,25,12,22);(7,9)(11,17,20,18,21,22,25,19)(12,23)(13,24)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 170);