Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17523 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C5×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,12)(10,14)(11,19)(13,17)(16,20)(18,21)(22,23,26,29,27,31,25,30)(24,32,34,28,35,36,33,37);(6,8)(7,13)(9,16)(10,17)(11,20)(12,18)(14,21)(15,19)(22,24)(23,28)(25,33)(26,34)(27,35)(29,36)(30,32)(31,37);(6,9,12,7)(8,16,18,13)(10,14,11,19)(15,17,21,20)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32);(6,10,12,11)(7,14,9,19)(8,17,18,20)(13,21,16,15)(22,25,27,26)(23,30,31,29)(24,33,35,34)(28,32,37,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10511);