Determining an upper bound for the exponent of the Schur multiplier and its generalization, the nilpotent multiplier, of a finite group are of interest. In 1974 M.R.Jones [2] attained an upper bound for the exponent of the Schur multiplier, and then in 1999 G.Ellis [1] determined a bound for nilpotent multiplier of some prime-power groups. The recent bound also improves the bound obtained by M.R.Jones. In 2000, the author in a joint paper [3] sharpened the bound introduced by G.Ellis in case of the Schur multiplier, and naturally the one obtained by M.R.Jones. In this article we intend to provide a smaller bound for the nilpotent multiplier of some prime-power groups.

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