Anhydrous $\mathrm{A}\mathrm{l}\mathrm{C}{\mathrm{l}}_3$ exists as $\mathrm{A}{\mathrm{l}}_2\mathrm{C}{\mathrm{l}}_6$ (dimer)


White fumes are due to the presence of $\mathrm{H}\mathrm{C}\mathrm{l}$ gas. Anhydrous aluminium chloride is hydrolysed partially with the moisture in the atmosphere to give $\mathrm{H}\mathrm{C}\mathrm{l}$ gas. This $\mathrm{H}\mathrm{C}\mathrm{l}$ combines with the moisture in the air and appears white in colour.
$\mathrm{A}\mathrm{l}\mathrm{C}{\mathrm{l}}_3\mathrm{}+3{\mathrm{H}}_2\mathrm{}\mathrm{O}\stackrel{\hspace{1em}\hspace{1em}}{\to }\mathrm{A}\mathrm{l}{\left(\mathrm{O}\mathrm{H}\right)}_3\mathrm{}+3\mathrm{H}\mathrm{C}\mathrm{l}\left(\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{ }\mathrm{f}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{s}\right)$

Thus ${\text{AlCl}}_{\text{3}}$ is a covalent compound rather than ionic. Fajan's rule states that, small size of cation, higher the charge on cation and large size of anion, greater is the covalent character of the ionic bond.