A note on some special $p$-groups

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 $d\ge 3$ and $p\mathrm{\ne }2.$ 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 $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }=p^{\frac{1}{2}d(d-1)-1}$ for $d\ge 3$ and $p\mathrm{\ne }2.$ Additionally, when $G$ is of exponent $p,$ we give the structure of $G\mathrm{.}$