Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #342 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C22×C22⋊C4 | C22×C5 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C22⋊C4 ◁ G | |
560 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,16,17,11)(9,13)(10,15)(12,18,20,21)(14,19);(7,11,13,12)(14,20,15,16)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 201);