Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20678 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1330 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)(10,11,15,20,16,21,14,19)(12,22,25,33,26,32,24,17)(13,27,28,18,30,34,29,35)(23,37,38,36,40,31,39,41);(10,12,14,24,16,26,15,25)(11,17,20,22,21,33,19,32)(13,23,28,38,30,40,29,39)(18,31,27,36,35,37,34,41);(7,9)(10,13)(11,18)(12,23)(14,28)(15,29)(16,30)(17,31)(19,34)(20,27)(21,35)(22,36)(24,38)(25,39)(26,40)(32,41)(33,37);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,28,30,29)(17,32,33,22)(18,34,35,27)(23,38,40,39)(31,41,37,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13666);