Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7523 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7523 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1404 |
| commutator | #125 | C2×C4×C5 |
| Frattini, cosocle | #490 | C22×C5 |
| prime radical | C5×#1656 | C2 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C4×C5 ◁ C42×C5 ◁ C5×#490 ◁ G | |
158 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 158 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)(8,14,16,11)(9,13)(10,12,17,21)(15,19,18,20);(7,11)(8,15)(10,17)(12,19)(13,20)(14,21)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 511);