P. Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) ,(3=2) log2 jGj when G is nilpotent. Using only Hall's theorem, this is the best one can do when jGj = 2. Using a result of G. J. Sherman, we improve the constant 3=2 to 5=3, which is best possible across all nilpotent groups and to 15=8 when G is nilpotent and jGj n 6= 8,16. These results are then used to prove that k(G) > log3 (jGj) when G=N is nilpotent, under natural conditions on N E G. Also, when G 0 is nilpotent of class c, we prove that k(G) ,(log jGj) t when jGj is large enough, depending only on (c,t).