July 20 Brain Strain: The Powers of Two Proposal

Beezer Wentwistal asked Zenobia Tightbottom (from the famous mathematical family of Tightbottoms) to be his wife.  She answered his proposal with one of her own:

"I will marry you if you can solve a math challenge.  I am thinking of a multi-digit integer N that is a power of 2.  Moreover, each digit in my integer N is a power of 2.  What is N?  If you cannot answer I'm afraid I cannot consider you a viable mate since I require intelligent offspring."

Can you help Beezer marry the love of his life?