What command can I use to find the name of the package manager used by my OS?
As far as I remember, there is no way to determine what package manager is used by the system through a command. Usually, the package manager used are system specific and there is no way to use an alternative of that unless they are a front-end or back-end of the package manager.
For example, Ubuntu uses dpkg. Software-center is a graphical front-end of Advanced Packaging Tool (APT), which in turn is a command line front-end of dpkg.
You can take a look at the available features of various package managers at DistroWatch.
Also, to know what package managers are used by various operating system, you can check out the list of software package management systems on Wikipedia.