We have computed a consistent power-commutator presentation for \(R(2,7)\), the largest finite two generator group of exponent \(7\). Our results show that \(R(2,7)\) is nilpotent of class 28 and has order \(7^{20416}\). This result is quite a surprise, as Lie algebra computations of the second author with M. F. Newman had led us to expect that the class of \(R(2,7)\) would be \(29\), and that the order would be \(7^{20418}\). So \(R(2,7)\) is the first known example of a Burnside group of prime exponent \(p\) which satisfies Lie relators which are not consequences of the multilinear Lie relators which hold in all groups of exponent \(p\).
We made several modifications to the version of the \(p\)-quotient algorithm available in the algebraic programming languages GAP and Magma, to enable us to complete our computations. But even with these modifications the computation took the best part of a year of CPU time, and \(1.5\) gigabytes of RAM. The most significant of our modifications was to make use of the Baker-Campbell-Hausdorff formula to construct part of the presentation of \(R(2,7)\).