Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
990 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)(6,7,9,8)(10,11)(12,21)(13,25)(14,19)(15,30)(16,20)(17,23)(18,28)(22,36)(24,32)(26,40)(27,34)(29,33)(31,41)(35,38)(37,39);(6,8,9,7)(10,12,15,23,16,24,14,29)(11,17,19,21,20,33,30,32)(13,22,27,37,28,38,26,41)(18,31,34,35,25,39,40,36);(10,13,15,27,16,28,14,26)(11,18,19,34,20,25,30,40)(12,22,23,37,24,38,29,41)(17,31,21,35,33,39,32,36);(10,14,16,15)(11,19,20,30)(12,23,24,29)(13,26,28,27)(17,32,33,21)(18,34,25,40)(22,37,38,41)(31,36,39,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14065);