Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7225 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | H(3,ℤ/4) |
| commutator | C2×C8 | C42×C5 |
| Frattini, cosocle | C4×M16 | C22×C5 |
| socle | C2×C5 | #1288 |
| prime radical | C23×C5 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ G | |
192 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 192 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)(8,16,18,11)(9,14)(10,15)(12,19,20,21)(13,17);(7,11,13,20,15,16,14,12)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 213);