Hello to all of you.

I would like to tell you about this weird problem I'm having when opperating with matrices  in mathematica.
I want to calculate the eigenssystem of a matrix which has seemingly not too high dimensions (729x729). The matrix is a quite good one (e.g. with many differnt symmetres such as SU(2);  Z(3) within 2 blocks etc.), but it seems that it takes forever to do that, at least more then 5-min (I did not wait that long). I fed the (NUMERICAl) matix directly to the function "Eigenvalues". On the other hand I found on the web an example kkkk = 1000;Timing[Eigenvalues[RandomReal[{0, 1}, {kkkk, kkkk}]]][[1]] -> 1.03 which is done very fast (I checked  that). 
So it would be  really helpful  if you could tell me: whats the problem here that with a matix which has no symmetries whatsoever the calculation is done in a moment, while a "better one is" just killing the computer? Does Mathematica have any  preferences regarding to exactly what kind of matrices should be fed in? 
Thank you in addvance and I'm sorry if it was an obvious one and just wasted your time