What is the basic idea?
Normally for classification, I use Sylow theorem to use semidirect product, but since $125 = 5^3$, I cannot use it.
All I know is that $Z(G)$ cannot be identity and from the cauchy theorem I necessarily have subgroup of order $5$, but I don't seem to use these facts to figure out the ways to classify group of order $125.$
Let $p$ be a prime number. Then there are, up to isomorphism, five groups of order $p^3$. These include three abelian groups and two non-abelian groups. The nature of the two non-abelian groups is somewhat different for the case $p = 2$.
The three abelian groups correspond to the three partitions of $3$.
The two non-abelian groups: For the case $p = 2$, these are dihedral group:$D_{8}$ and quaternion group. For the case of odd $p$, these are unitriangular matrix group:$UT(3,p)$ and semidirect product of cyclic group of prime-square order and cyclic group of prime order.