Burnside proved in [1] that the order of $B(2,4)$ is $\le4096$. Tobin proved in [4] that this number is the correct order for this group. A short presentation for $B(2,4)$ was given by Leech in [3], which is reproduced in Coxeter-Moser's book [2]:
$$
B(2,4)=\langle a,b \mid a^4, b^4, (ab)^4, (a^{-1}b)^4, (ab^2)^4, (a^2b)^4, (a^{-1}b^{-1}ab)^4, (a^2b^2)^4, (a^{-1}bab)^4, (ab^{-1}ab)^4\rangle
$$
Any one of the last four relators can be omitted.

UPDATE 5/23/23:
Interestingly, the first 6 relators are of the form $w^4$, where $w$ is some primitive element of the free group on $a,b$ (which means that $w$ is part of some basis of the free group $F(a,b)$). Whereas the other 4 relators are of the form $u^4$ where $u$ is not primitive.

References:

[1] Burnside, W. On an unsettled question in the theory of discontinuous groups. Quart. J. 33, 230-238 (1902).

[2] Coxeter, H. S. M.; Moser, W. O. J. Generators and relations for discrete groups.
Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980.

[3] Leech, John. Coset enumeration on digital computers. Proc. Cambridge Philos. Soc. 59 (1963), 257–267.

[4] Tobin, John Joseph. On Groups with Exponent Four. Thesis (Ph.D.)–The University of Manchester (United Kingdom). 1954. 107 pp.