Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7310 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#7310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1304 |
| commutator | C24 | C42×C5 |
| Frattini, cosocle | C23×D4 | C22×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C23×C5×D4 ◁ G | |
650 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,9,13)(8,11,14,18)(10,12,17,20)(15,19,16,21);(6,8)(7,11,12,19)(9,16,17,14)(10,15)(13,18)(20,21)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 298);