I just installed Centos 6.2 basic server. During installation I have created partition for / and swap. Now linux is starting with command prompt with logon and password. I have used yum groupinstall -Y 'X window System' shell replying something like that couldn't get list from mirror. I put computer name as localhost.localdomain.

Can anybody show me how I can enable GUI for Centos 6.2 basic server install.

Salaam,
What is the output of the yum check-update command? Have you been able to get network connectivity. Try ping www.BBC.co.UK.

Almost ab definitio a basic server install is one without a GUI. So, what do you really want, a GUI or a basic server install?

The X window system is probably not really what you want; X Windows underlies many popular GUIs but isn't a desktop or a window manager itself. If you wanted, say, XFCE and told yum to install that, it would automagically get X windows for you from the network, if it wasn't already available.

It sounds, however, as if something fairly fundamental in networking isn't set up and working, and without curing that yum isn't going to be able to get much of anything for you from the 'net. How do you connect, and can you ping sites on the internet?

As Salasi said, it sounds like your network isn't connected.
Try a yum cmd again and post the exact cmd and output msgs for us to see.
Also, (as he said) choose a desktop env eg Gnome and yum will (if it works) pull down dependencies like X-Win automatically.