Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #63093⋊C5 |
| Fitting, Sylow | #63093 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C24⋊C5 |
| commutator, cosocle | #585 | C22×C5 |
| Frattini | C4 | C22×C24⋊C5 |
| Fitting | #63093 | C5 |
| hypercenter | D8 | C24⋊C5 |
| hypocenter | 2−1+4 | C5×D4 |
| LCS | G ▷ #585 ▷ 2−1+4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
1104 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
5,2^8|b^2=i,c^2=i,h^2=i,b^a=bgi,c^a=cgi,d^a=dfghi,e^a=defhi,f^a=dghi,g^a=degh,c^b=ch,h^b=hi,h^c=hi,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116346);