Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #106 |
| 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)(6,7,10,16,12,18,28,33,13,19,27,15,11,17,9,26)(8,20,22,31,24,36,21,32,25,34,29,30,23,35,37,14);(6,8)(7,14,17,30,19,32,18,31)(9,21)(10,29)(11,23)(12,24)(13,25)(15,20,33,35,16,34,26,36)(22,27)(28,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 168);