Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11837 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1514 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| prime radical | C5×#3184 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×M16 ◁ G | |
324 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 324 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)(8,16,18,11)(9,12)(10,15)(13,17,20,21)(14,19);(7,11)(8,17)(9,19)(12,20)(13,14)(15,16)(18,21);(7,12,15,14)(8,18)(11,20,16,13)(17,21)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4825);