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Shephali SwarnkarShephali Swarnkar 

access permission to user/profile

How to give access permission to user/profile on the field  'LastActivityDate' as i can not see this field name in field permission table(setup ->manage user -> profiles ->Object Settings->field permission
Akhil AnilAkhil Anil
LastActivityDate will be available if the "Allow activities" checkbox under Optional Features is ticked in the custom object's setup page. You will not see this field in the field permission table as it is a System field and is automatically updated.
Himanshu ParasharHimanshu Parashar
Hi Shephali,

LastActivityDate is only available if the "Allow activities" checkbox under Optional Features is ticked in the custom object's setup page. If it is not checked, this system field will not be available through the API. 

Here is the SFDC article for the same.

https://help.salesforce.com/apex/HTViewSolution?id=000181816&language=en_US

Thanks,
Himanshu
Shephali SwarnkarShephali Swarnkar
Hi Akhil and Himanshu,
      Thanks to both for reply.
Can i have the navigation to check the above,coz i am unable to find it out.
Akhil AnilAkhil Anil
Below is the navigation

Setup --> Build --> Create --> Objects --> Click corresponding object --> Click Edit --> You should see the "Allow Activities" checkbox in the "Optional Features" section below. 
Shephali SwarnkarShephali Swarnkar
Thanks Akhil for your reply.
 I want this fileld for contact object(standard object).
Akhil AnilAkhil Anil
Go to Contacts (tab) --> Click on Edit Layout (top right) --> Click Related Lists (top left pane) --> Drag and Drop Activity History to one of the sections below. You should then see the Activity related list for your Contact records.

Kindly mark this as an Answer if you find this useful.