Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13579 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#353 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | #124 | C23×C5 |
| prime radical | C5×#3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C5×#532 ◁ G | |
454 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 21 points:
(1,2,3,4,5)(6,7,9,14)(8,12,16,19)(10,13,17,20)(11,15,18,21);(7,12)(8,10)(9,17)(13,15)(16,18)(19,20);(6,8,11,10)(7,12,15,13)(9,16,18,17)(14,19,21,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6567);