Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
456 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)(6,7,12,17,11,16,30,34,13,18,32,36,10,29,20,21)(8,19,23,37,24,9,26,28,25,27,15,35,22,14,33,31);(6,8,20,33,10,22,32,15,13,25,30,26,11,24,12,23)(7,14,34,37,29,27,17,31,18,9,21,35,16,19,36,28);(6,9)(7,15)(8,21)(10,19)(11,27)(12,31)(13,14)(16,33)(17,24)(18,23)(20,35)(22,36)(25,34)(26,29)(28,32)(30,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5974);