Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10753 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11,14,20,16,21,15,19)(12,22,26,31,25,32,24,17)(13,18,28,33,29,35,27,34)(23,37,38,30,39,36,40,41);(10,12,15,24,16,25,14,26)(11,17,19,32,21,31,20,22)(13,23,27,40,29,39,28,38)(18,30,34,37,35,41,33,36);(10,13)(11,18)(12,23)(14,28)(15,27)(16,29)(17,30)(19,34)(20,33)(21,35)(22,36)(24,40)(25,39)(26,38)(31,41)(32,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3741);