Is it permissible in Islam to go for second marriage without informing anybody or even not informing the first wife who is still alive and staying with the husband who is trying to seek second wife.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The husband is not required to take the permission of his wife or tell her about his getting married to a second wife.
The husband is also not asked to inform people other than the witnesses and the Wali (guardian of the woman) about his marriage. But there must be a Wali and two witnesses in the marriage contract. He may tell these three people to keep this marriage secret. But the most preponderant view is that he should make the marriage public.

Allah knows best.