Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20852 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×#41 ◁ G | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,19,16,20,14,28)(12,21,23,17,24,31,29,32)(13,18,27,34,25,33,26,38)(22,35,37,30,36,41,39,40);(10,12,15,23,16,24,14,29)(11,17,19,31,20,32,28,21)(13,22,27,37,25,36,26,39)(18,30,34,41,33,40,38,35);(7,9)(10,13)(11,18)(12,22)(14,26)(15,27)(16,25)(17,30)(19,34)(20,33)(21,35)(23,37)(24,36)(28,38)(29,39)(31,41)(32,40);(13,25)(18,33)(22,36)(26,27)(30,40)(34,38)(35,41)(37,39)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13840);