Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20543 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19,21,24,23,25,22,20);(11,14)(12,16)(18,20,22,25,23,24,21,19);(10,12,13,16)(11,15,14,17);(11,14)(15,17)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13531);