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×#22892 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
986 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)(10,11,17,23)(12,24,13,27)(14,20,32,38)(15,22,16,21)(18,29,19,26)(25,40,28,36)(30,35,41,34)(31,39,33,37);(6,8,9,7)(10,12,17,13)(11,18,23,19)(14,30,32,41)(15,29,16,26)(20,36,38,40)(21,27,22,24)(25,33,28,31)(34,39,35,37);(10,13,17,12)(11,19,23,18)(14,28,32,25)(15,29,16,26)(20,35,38,34)(21,27,22,24)(30,31,41,33)(36,37,40,39);(10,14)(11,20)(12,25)(13,28)(15,33)(16,31)(17,32)(18,34)(19,35)(21,39)(22,37)(23,38)(24,40)(26,41)(27,36)(29,30)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15880);