Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23907 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1942 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)(10,11,14,17)(12,18,19,25)(13,21,22,16)(15,23,24,20);(6,7)(8,9)(13,22)(16,21)(18,25)(20,23);(7,9)(10,12)(11,15)(13,23)(14,19)(16,25)(17,24)(18,21)(20,22);(13,22)(16,21)(18,25)(20,23)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16895);