Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C16 · Schur multiplier (H2) = C4 |
| direct product | C5×#7053 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C42 |
| commutator | C4 | C4×C5×C16 |
| Frattini, cosocle | C82 | C22×C5 |
| socle, prime radical | C22×C5 | #322 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,8,12,9,14,16,20,10,13,15,21,17,11,18,19);(7,11,13,14)(8,15)(12,20,21,19)(16,18)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 41);