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×#22898 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2358 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
996 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)(6,7,10,16,11,17,26,15)(8,14,20,29,21,28,19,12)(9,22,23,32,25,31,24,13)(18,33,35,37,36,27,34,30);(6,8,10,20,11,21,26,19)(7,12,16,14,17,29,15,28)(9,18,23,35,25,36,24,34)(13,27,22,30,32,33,31,37);(6,9)(7,13)(8,18)(10,23)(11,25)(12,27)(14,30)(15,31)(16,22)(17,32)(19,34)(20,35)(21,36)(24,26)(28,37)(29,33);(6,8)(7,14)(9,18)(10,20)(11,21)(12,15)(13,30)(16,29)(17,28)(19,26)(22,33)(23,35)(24,34)(25,36)(27,31)(32,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15886);