Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
840 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)(10,11,15,20,16,21,14,19)(12,22,25,17,26,33,24,34)(13,27,30,36,31,18,29,37)(23,38,39,41,40,35,28,32);(6,8,9,7)(10,12)(11,17)(13,28)(14,25)(15,24)(16,26)(18,35)(19,33)(20,22)(21,34)(23,29)(27,38)(30,40)(31,39)(32,36)(37,41);(10,12)(11,17)(13,23)(14,24)(15,25)(16,26)(18,32)(19,22)(20,33)(21,34)(27,41)(28,29)(30,39)(31,40)(35,36)(37,38);(10,13,14,29,16,31,15,30)(11,18,19,36,21,27,20,37)(12,23,24,28,26,40,25,39)(17,32,22,35,34,41,33,38)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16035);