Burnside asked questions about periodic groups in his influential paper of
1902. The study of groups with exponent six is a special case of the study
of the Burnside questions on which there has been significant progress. It
has contributed a number of worthwhile aspects to the theory of groups and
in particular to computation related to groups. Finitely generated groups w
ith exponent six are finite. We investigate the nature of relations require
d to provide proofs of finiteness for some groups with exponent six. We giv
e upper and lower bounds for the number of sixth powers needed to define th
e largest 2-generator group with exponent six. We solve related questions a
bout other groups with exponent sis using substantial computations which we
explain.