Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#7179 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | D8 |
| 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×C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
274 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)(10,11,13,19,15,20,22,18,16,21,12,17,14,24,23,25);(7,9)(11,17,20,25,21,19,24,18)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 167);