Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×#329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C23×C5×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
1502 conjugacy classes of subgroups; only the 462 normal subgroups are shown.
This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 462 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(21,22,23,24,25);(3,4);(5,6,7,10,8,9,11,12)(14,17,18,19);(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10621);