Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7440 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#7440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1353 |
| commutator | C2×M16 | C2×C4×C5 |
| Frattini, cosocle | #423 | C22×C5 |
| prime radical | C23×C5 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C5×#423 ◁ G | |
144 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 42 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)(7,11,12,18,13,19,20,15)(8,14)(9,16);(6,7)(8,15,14,18)(9,12)(10,13)(11,17,19,21)(16,20)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 428);