Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1390 |
| commutator | C2×D4 | C42×C5 |
| Frattini, cosocle | C2×#127 | C22×C5 |
| prime radical | C2×C5×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C4×C5 ◁ C2×C5×#127 ◁ G | |
334 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 66 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)(8,15,17,21)(9,14,19,12)(10,13)(11,16,20,18);(6,8)(7,11,13,21)(9,18)(10,16)(12,20,14,15)(17,19)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 481);