Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(11,17)(18,21)(22,23,27,31)(24,32,35,45)(25,29,36,43)(26,38,40,30)(28,39,42,34)(33,46,47,53)(37,50,49,44)(41,51,52,48);(6,7)(8,11)(9,12)(10,13)(14,18)(15,17)(16,19)(20,21)(22,24,27,35)(23,28,31,42)(25,33,36,47)(26,39,40,34)(29,41,43,52)(30,45,38,32)(37,51,49,48)(44,53,50,46);(6,8)(7,11)(9,14)(10,15)(12,18)(13,17)(16,20)(19,21)(22,25,27,36)(23,29,31,43)(24,33,35,47)(26,37,40,49)(28,41,42,52)(30,44,38,50)(32,46,45,53)(34,48,39,51);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,18,17,21)(25,36)(29,43)(33,47)(37,49)(41,52)(44,50)(46,53)(48,51)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11235);