  Hexagonal superlattice formed by sodium multi-vacancy cluster ordering in
Na$_{0.77}$CoO$_2$ has been proposed based on synchrotron X-ray Laue
diffraction study on electrochemically fine-tuned single crystals. The title
compound sits closely to the proposed lower end of the miscibility gap of x ~
0.77-0.82 phase separated range. The average sodium vacancy cluster size is
estimated to be 4.5 Na vacancies per layer within a large superlattice size of
sqrt{19}a*sqrt{19}a*3c. The exceptionally large Na vacancy cluster size favors
large twinned simple hexagonal superlattice of sqrt{19}a, in competition with
the smaller di-, tri- and quadri-vacancy clusters formed superlattices of
sqrt{12}a and sqrt{13}a. Competing electronic correlations are revealed by the
observed spin glass-like magnetic hysteresis below ~ 3K and the twin, triple
and mono domain transformations during thermal cycling between 273-373K.
