Classification of GVZ and Nested GVZ $p$-groups up to Order $p^6$

Let $G$ be a finite group and let $\text{Irr}(G)$ denote the set of irreducible complex characters of $G$. For a normal subgroup $N⊴G$ and $\chi \in \text{Irr}(G)$, we say that $\chi$ is \emph{fully ramified} over $N$ if $\chi (g)=0$ for all $g\in G\setminus N$. A group $G$ is said to be of \emph{central type} if there exists $\chi \in \text{Irr}(G)$ that is fully ramified over $Z(G)$. Motivated by this notion, an irreducible character $\chi \in \text{Irr}(G)$ is called of \emph{central type} if $\chi$ vanishes on $G\setminus Z(\chi )$, where $$Z(\chi )=\{g\in G:\mathrm{\mid }\chi (g)\mathrm{\mid }=\chi (1)\}$$ is the center of $\chi$. Groups in which every irreducible character is of central type are called \emph{GVZ-groups}. Furthermore, a group $G$ is said to be \emph{nested} if for all $\chi ,\psi \in \text{Irr}(G)$, either $Z(\chi )\subseteq Z(\psi )$ or $Z(\psi )\subseteq Z(\chi )$. It is known that a GVZ-group is nilpotent. In this article, we classify all GVZ and nested GVZ $p$-groups of order at most $p^6$, where $p$ is an odd prime.