Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1993 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,19,23,35)(9,25,28,33)(10,29)(11,16)(12,18)(13,14,27,32)(15,20)(17,31)(21,30)(22,26,24,37)(34,36);(6,8,20,33,31,22,10,14,12,23,36,25,11,24,30,32)(7,13,21,37,16,28,34,35,18,27,29,26,17,9,15,19);(6,9,12,28)(7,14)(8,21)(10,26,30,37)(11,27,31,13)(15,24)(16,25)(17,33)(18,32)(19,20,35,36)(22,34)(23,29)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6279);