Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C22×C5×#448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C22×C5×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C24×C4×C5 ◁ G | |
5326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(17,18,19,20,21);(3,4);(5,6)(7,8)(10,13)(11,15)(14,16);(9,10,12,14)(11,16,15,13);(6,8)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53337);