Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1402 |
| commutator | #127 | C2×C4×C5 |
| Frattini, cosocle | #493 | C22×C5 |
| prime radical | C5×#493 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C4×C5 ◁ C42×C5 ◁ C5×#493 ◁ G | |
162 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7)(8,13,16,21)(9,17,18,20)(10,14)(11,15,12,19);(6,8,10,16)(7,11,14,17)(9,15,19,18)(12,13,20,21)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 504);