Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19)(22,23,27,33,29,32,42,50,28,34,41,51,26,43,25,31)(24,35,38,49,37,30,36,46,39,48,52,45,44,47,40,53);(6,8,10,16)(7,11,13,14)(9,15,17,21)(12,18,19,20)(22,24,28,39)(23,30,34,47)(25,40,42,36)(26,44,29,37)(27,38,41,52)(31,49,50,45)(32,48,43,35)(33,46,51,53);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,25,26,41,28,42,29,27)(23,31,43,51,34,50,32,33)(24,36,44,38,39,40,37,52)(30,45,35,46,47,49,48,53)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4486);