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Let $G$ be a finitely generate nilpotent class-2 torsion-free group. We studyhow the zeta function enumerating normal subgroups of G varies on residueclasses. In particular, we show that for small such $G$ of Hirsch length lessthan or equal to 7, the normal zeta functions are generically always rationalfunctions on residue classes. We then show that there are examples of groupswith Hirsch length 8 whose normal zeta function is not a rational function onresidue classes. We observe the connection to Higman's PORC conjecture.

Normal zeta functions of small $\mathfrak{T}_2$-groups and their behaviour on residue classes