A Lie algebra is called a generalized Heisenberg algebra of degree n if its centre coincides with its derived algebra and is n-dimensional. In this paper we define for each positive integer n a generalized Heisenberg algebra 𝓗ₙ. We show that 𝓗ₙ and 𝓗 ₁ⁿ, the Lie algebra which is the direct product of n copies of 𝓗 ₁, contain isomorphic copies of each other. We show that 𝓗ₙ is an indecomposable Lie algebra. We prove that 𝓗ₙ and 𝓗 ₁ⁿ are not quotients of each other when n ≥ 2, but 𝓗 ₁ is a quotient of 𝓗ₙ for each positive integer n. These results are used to obtain several families of 𝓗ₙ-modules from the Fock space representation of 𝓗 ₁. Analogues of Verma modules for 𝓗ₙ, n ≥ 2, are also constructed using the set of rational primes.