We know that mapping class group (MCG) $\Gamma_1$ for genus 1 closed surface generated by to elements: $U$ of order 6 and $S$ of order 4. There is defining relation that totally fixed the MCG $\Gamma_1$: $U^3=S^2$. (Is this correct?)

In arXiv:math/0309299, Korkmaz showed that mapping class group $\Gamma_2$ for genus 2 closed surface is also generated by to elements: $A_4A_3A_2A_1$ of order 10 and $A_5A_4A_3A_2A_1$ of order 6.

Do we know the defining relations among the two generator that define $\Gamma_2$?

We know that mapping class group (MCG) $\Gamma_1$ for genus 1 closed surface is generated by two elements: $U$ of order 6 and $S$ of order 4. There is a defining relation that totally fixed the MCG $\Gamma_1$: $U^3=S^2$. (Is this correct?)

In arXiv:math/0309299, Korkmaz showed that mapping class group $\Gamma_2$ for genus 2 closed surface is also generated by two elements: $U_1=A_4A_3A_2A_1$ of order 10 and $U_2=A_5A_4A_3A_2A_1$ of order 6.

Do we know the defining relations among the two generators $U_1$ and $U_2$ that will allow us to define $\Gamma_2$?