Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20863 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,20)(13,22)(14,17)(15,18)(16,19)(21,24)(23,25);(6,8,9,7)(10,12)(11,13)(14,21)(15,22)(16,23)(17,20)(18,25)(19,24);(10,13)(11,12)(14,20)(15,25)(16,24)(17,21)(18,22)(19,23);(10,14,16,15)(11,17,19,18)(12,21,23,22)(13,20,24,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13851);