  Recently Rai obtained an upper bound for the order of the Schur multiplier of
a $d$-generator special $p$-group when its derived subgroup has the maximum
value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to
obtain the Schur multiplier, the exterior square and the tensor square of such
$p$-groups. Then we specify which ones are capable. Moreover, we give an upper
bound for the order of the Schur multiplier, the exterior product and the
tensor square of a $d$-generator special $p$-group $ G $ when $
|G'|=p^{\frac{1}{2}d(d-1)-1}$ for $ d\geq 3 $ and $ p\neq 2. $ Additionally,
when $ G $ is of exponent $ p, $ we give the structure of $ G. $
