Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22695 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | D4×C4⋊C4 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,18,13,19,12,17)(8,20,24,32,25,36,23,14)(9,26,27,34,28,31,22,15)(10,16,30,35,21,33,29,37);(6,8)(7,14)(9,10)(11,24)(12,23)(13,25)(15,33)(16,34)(17,36)(18,20)(19,32)(21,28)(22,29)(26,37)(27,30)(31,35);(6,9,12,22,13,28,11,27)(7,15,18,26,19,34,17,31)(8,21,23,30,25,10,24,29)(14,16,20,35,32,33,36,37);(6,10)(7,16)(8,22)(9,24)(11,30)(12,29)(13,21)(14,31)(15,20)(17,37)(18,35)(19,33)(23,28)(25,27)(26,32)(34,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15683);