Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#8937 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C23×C5 | C2×C22⋊C4 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(11,14,15,16)(18,19,21,23,22,25,20,24);(10,11,12,16,13,15,17,14)(19,23)(20,21)(24,25);(7,9)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1925);