Center of an upper unitriangular group ($Z(U_n(q))=\{I+t{E}_{1n}:t\in {\mathbb{F}}_q\},n\ge 3$)

The upper unitriangular group is generated by adjacent elementary transvection matrices $I+s{E}_{i,i+1}$. Commuting a matrix $X$ with each such generator forces every entry above the diagonal to vanish except possibly its $(1,n)$ entry: comparing $X{E}_{i,i+1}$ with $E_{i,i+1}X$ kills the entries above position $(i,i)$ in column $i$ and to the right of position $(i+1,i+1)$ in row $i+1$. The surviving matrices $I+t{E}_{1n}$ do commute with the entire group. In particular the group center has order $q$; it distinguishes $U_4(2)$ from $U_3(4)$ despite their common order $64$.