Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 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)(15,20,17,19)(18,21)(22,23,27,33,29,35,28,34)(24,36,39,51,41,52,40,47)(25,42,26,45,44,32,43,31)(30,46,48,38,50,37,49,53);(6,8)(7,11)(9,17)(10,16)(12,20)(13,14)(15,18)(19,21)(22,24)(23,30)(25,38)(26,46)(27,40)(28,39)(29,41)(31,36)(32,51)(33,49)(34,48)(35,50)(37,43)(42,47)(44,53)(45,52);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,25,28,43,29,44,27,26)(23,31,33,42,35,45,34,32)(24,37,40,38,41,46,39,53)(30,47,48,36,50,51,49,52)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4854);