Is it true that a set of generators for $PSL(3,2)\simeq SL(3,2)$ is: $$\alpha=\left(\begin{array}{ccccccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array}\right),$$ $$\beta=\left(\begin{array}{ccccccc} 1&0&0\\ 0&0&1\\ 0&1&0\\ \end{array}\right),$$
$$\gamma=\left(\begin{array}{ccccccc} 1&0&0\\ 0&1&0\\ 0&1&1\\ \end{array}\right)?$$ How can I check/prove this fact? Thanks!
Start with any matrix $A\in PSL(3,2)$ and use elementary row operations (Gaussian elimination) to reduce $A$ to the identity matrix. Each elementary operation can be accomplished by multiplying on the left by one of the following nine elementary matrices: $$ E_{12}=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\; E_{13}=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\; E_{23}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix},\; $$
$$ E_{21}=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\; E_{31}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{pmatrix},\; E_{32}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1\end{pmatrix},\; $$
$$ R_{12}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix},\; R_{13}=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{pmatrix},\; R_{23}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}. $$
Now noting that each elementary matrix equals its own inverse, we see that $PSL(3, 2)$ is generated by the nine elementary matrices listed above. In fact, many of these matrices are redundant, and the set $\{R_{13}, R_{23}, E_{32}\}$, or $\{E_{23}, R_{12}, R_{23}\}$ already generates the whole group. This follows from formulas like $R_{13} = R_{12}R_{23}R_{12}$, $E_{13} = R_{12}E_{23}R_{12}$, etc. Indeed, the minimal number of generators is even equal to two, see here.