Abelian groups of order 9, and groups of order 27

For order \[ 9=3^2, \] the fundamental theorem of finite abelian groups gives exactly \[ \boxed{C_9,\qquad C_3\times C_3}. \] They are not isomorphic because \(C_9\) has an element of order \(9\), whereas every nonidentity element of \(C_3^2\) has order \(3\).

For order \[ 27=3^3, \] the abelian groups are \[ \boxed{C_{27},\qquad C_9\times C_3,\qquad C_3^3}. \] Their exponents are respectively \(27,9,3\), so they are pairwise nonisomorphic.

There are exactly two nonabelian groups of order \(27\). One is the Heisenberg group \[ UT_3(\mathbb F_3) = \left\{ \begin{pmatrix} 1&a&b\\0&1&c\\0&0&1 \end{pmatrix} :a,b,c\in\mathbb F_3 \right\}, \] which has exponent \(3\). The other is the nontrivial semidirect product \[ C_9\rtimes C_3 =\langle x,y\mid x^9=y^3=1,\ yxy^{-1}=x^4\rangle, \] which contains elements of order \(9\). Hence these two are nonisomorphic.

Thus there are five groups of order \(27\) up to isomorphism: three abelian and two nonabelian.