Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20867 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#385 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1312 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)(6,7,11,16,12,15,10,26)(8,17,19,13,21,29,20,28)(9,22,24,14,25,31,23,30)(18,33,34,32,35,27,36,37);(6,8,11,19,12,21,10,20)(7,13,16,29,15,28,26,17)(9,18,24,34,25,35,23,36)(14,27,31,37,30,33,22,32);(6,9)(7,14)(8,18)(10,23)(11,24)(12,25)(13,27)(15,30)(16,31)(17,32)(19,34)(20,36)(21,35)(22,26)(28,33)(29,37);(7,15)(13,28)(14,30)(16,26)(17,29)(22,31)(27,33)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13855);