Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8933 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C23×C5 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,11)(12,13)(14,15,18,23,20,17,19,22)(16,24,27,25,28,29,26,21);(6,7,9,10)(8,11,12,13)(14,16,19,26,20,28,18,27)(15,21,22,29,17,25,23,24);(6,8,9,12)(7,11,10,13)(14,17,20,15)(16,25,28,21)(18,22,19,23)(24,27,29,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1921);