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×#7181 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #102 |
| commutator | C8 | C4×C5×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| socle | C2×C5 | #1279 |
| prime radical | C23×C5 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,12,17,28,33,13,18,27,34,11,16,9,25)(8,19,21,30,23,36,20,14,24,29,37,31,22,35,26,32);(6,8,11,22,13,24,12,23)(7,14)(9,26,27,37,28,20,10,21)(15,29)(16,30)(17,31)(18,32)(19,34)(25,36)(33,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 169);