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×#17526 |
| 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×#3483 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
990 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)(8,14)(9,18)(10,15)(11,20)(12,16)(13,17)(19,21)(22,23,26,29,27,31,25,30)(24,32,34,28,35,36,33,37);(6,8,12,9)(7,13,16,21)(10,17,11,19)(14,15,18,20)(22,24,27,35)(23,28,31,37)(25,33,26,34)(29,36,30,32);(6,9,12,8)(7,14,16,18)(10,17,11,19)(13,15,21,20)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32);(6,10,12,11)(7,15,16,20)(8,17,9,19)(13,18,21,14)(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, 10514);