The Weibull distribution can be obtained using a power transformation from the standard exponential distribution. In this article, we will consider a symmetrized power transformation of a random variable with the standard normal distribution. We will call its distribution the $α$-{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of $α$-normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of $α$-normal random variables. Moreover, we define the joint distribution function of the multivariate $α$-normal distribution as a meta-Gaussian distribution with $α$-normal marginals. We consider also the limiting distribution as $α$ tends to infinity.