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×#23049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11,16,21,17,13,15,20)(12,22,19,18,25,14,24,23);(6,8,9,7)(10,12)(11,18)(13,23)(14,20)(15,19)(16,24)(17,25)(21,22);(10,13)(11,17)(12,23)(14,24)(15,21)(16,20)(18,25)(19,22);(10,14,16,23,17,22,15,18)(11,19,21,25,13,24,20,12)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16037);