An Interesting Matrix Modulo 27 Problem

This is a Hill Cipher implementation with 0=space, 1=a, 2=b, ... We have a message p that is encoded as c=pK where K is a 2x2 matrix with integer entries from 0 to 26 and both c and p are 1x2 vectors. The entries of c are reduced modulo 27. So p=cK^-1 where K^-1 is the inverse, modulo 27, of the matrix K. So I've found that the 2 most common vectors in the coded message are UJ and HF (which correspond to [21 10] and [8 6] respectively) and that these correspond to the uncoded message through parts of the word "the" and spaces before and after the word (ie. "space T"=[0 20], "TH"=[20 8], "HE"=[8 5], and "E space"=[5 0]). I'm not sure which these two specifically correspond to but I've tried all combinations in solving for K and can't seem to get a K that has all integer entries. Can anyone find this K or its inverse?