Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#12686 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #379 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1779 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
904 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)(6,7,10,16,11,17,26,34,12,14,25,30,9,15,23,32)(8,18,20,35,19,28,36,33,21,31,24,37,22,13,27,29);(6,8)(7,13)(9,22)(10,24)(11,19)(12,21)(14,28)(15,31)(16,33)(17,18)(20,25)(23,36)(26,27)(29,30)(32,35)(34,37);(7,14)(10,25)(13,28)(15,17)(18,31)(20,24)(23,26)(27,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5674);