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×#17524 |
| 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)(7,11)(8,14)(9,16)(12,19)(13,17)(18,20)(22,23,26,30,27,29,25,34)(24,31,32,28,33,37,35,36);(6,7,10,13)(8,12,14,20)(9,17,16,11)(15,19,21,18)(22,24,27,33)(23,28,29,36)(25,35,26,32)(30,37,34,31);(6,8)(7,12)(9,15)(10,14)(11,18)(13,20)(16,21)(17,19)(23,29)(28,36)(30,34)(31,37);(8,14)(12,20)(15,21)(18,19)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10512);