Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C8 |
| direct product | C5×#7529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1408 |
| commutator | C4×C8 | C2×C4×C5 |
| Frattini, cosocle | #493 | C22×C5 |
| prime radical | C5×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C4×C5 ◁ C42×C5 ◁ C5×#493 ◁ G | |
186 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 186 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)(8,13,16,12)(9,14,10,19)(11,15)(17,21,18,20);(7,12)(8,16)(9,10)(13,14)(15,21)(17,18)(19,20)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 517);