There is a book around 1968-71 which is a conference proceedings of a Conference in Oxford (maybe "Oxford Symposium on Finite Group Theory", edited by M.B. Powell and G. Higman), in which an article by Higman himself outlines how to characterize the alternating group $A_{n}$ by its character table. In it, a result implicit in Brauer's development of modular representation theory, they note the following result: let $R$ be a ring of algebraic integers containing all entries of the character table of a finite group $G$. Let $p$ be a rational prime divisor of $|G|$, and let $\pi$ be a prime ideal of $R$ containing $p$. Then two elements $x,y \in G$ have $G$- conjugate $p^{\prime}$-parts if and only if $\chi(x) \equiv \chi(y)$ (mod $\pi$) for each irreducible character in $G$. I won't give here the orthogonality relations from modular character theory which imply this result. Recall that we may write uniquely $x = uv = vu$ where the order of $u$ is a power of $p$ and the order of $v$ is prime to $p$. The element $v$ is known as the $p^{\prime}$ (or sometimes $p$-regular) part of $x$.
Applying this with $p =2$, for example, we see that $g_{2}$ has the same $2^{\prime}$-part as $g_{1}= e$, so that the order of $g_{2}$ is a power of $2$.
Similarly, the order of $g_{3}$ is a power of $2$. Likewise, the order of $g_{4}$ is a power of $3$. As for $g_{5}$ and $g_{6}$, this depends of course what $\alpha$ is. You would normally need to be told this. However, I think you can see from the other character values that $g_{5}$ and $g_{6}$ have to have order a power of $7$. With a little more thought, $g_{4}$ has order $3$. Looking at centralizer orders (there must be an involution somewhere!), we see that $g_{2}$ has order $2$. Looking at centralizer orders again, the Sylow $2$-subgroup is non-Abelian of order $8$ and $g_{3}$ must have order $4$. In fact, $g_{5}$ and $g_{6}$ each have order $7$.
(I should also have noted above that continuing the above argument, Higman notes that it is possible to determine the prime divisors of the order of each element of $G$ from its character table).
Actually, as noted in the comments below, this argument is "overkill"for this particular problem, though this more general argument has its own interest.
