Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#11493 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C22×C5×C16 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C23×C5 | C22×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19)(22,23,25,29,26,30,34,37,27,31,24,28,32,35,33,36);(6,8,10,16)(7,11,13,14)(9,15,17,21)(12,18,19,20);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,24,26,33,27,25,32,34)(23,28,30,36,31,29,35,37)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4481);