Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C5 · Schur multiplier (H2) = C28×C42 |
| direct product | C22×C5×#376 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4×C5 | C22 |
| commutator | C2 | C23×C42×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| socle, prime radical | C25×C5 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4×C5 ◁ G | |
5150 conjugacy classes of subgroups; only the 3614 normal subgroups are shown.
This group has 5150 conjugacy classes of subgroups — too many to draw. Only its 3614 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) = 1280 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(21,22,23,24,25);(3,4);(5,6,7,8)(13,14)(15,18)(16,17)(19,20);(9,10,11,12)(14,17)(18,20);(13,15,16,19)(14,18,17,20)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26975);