Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C8×C16 · Schur multiplier (H2) = C8 |
| direct product | C5×#7052 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C82 | C22 |
| commutator | C2 | C5×C8×C16 |
| Frattini, cosocle | C82 | C22×C5 |
| socle, prime radical | C22×C5 | C82 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C82 ◁ G | |
150 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)(6,7,8,12,9,13,14,19,10,11,15,17,16,18,20,21);(7,11)(12,17)(13,18)(19,21)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 40);