Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C5 · Schur multiplier (H2) = C216 |
| direct product | C24×C5×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26×C5 | C22 |
| commutator | C2 | C25×C4×C5 |
| Frattini, cosocle | C22 | C26×C5 |
| prime radical | C27×C5 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C26×C5 ◁ G | |
43678 conjugacy classes of subgroups; only the 15454 normal subgroups are shown.
This group has 43678 conjugacy classes of subgroups — too many to draw. Only its 15454 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) = 640 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2)(17,18,19,20,21);(3,4);(5,6);(7,8);(9,10,12,14)(11,13,15,16);(10,13)(14,16)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55627);