TABLE I: Critical properties of the Potts model on the 3-12 lattice. (H = Homogeneity Assump- tion, N = Numerical, T = Theoretical universality prediction.) $^{TABLE II: Critical properties of the Potts model on the kagome (1}$×1) lattice. (H = Homogeneity Assumption, N = Numerical, T = Theoretical universality prediction.) of the numerical accuracy of the finite-size analysis we can reach at present. For q = 2, the critical points obtained from (6) under the homogeneity assumption coincide with our numerical estimations to 5 or so decimal places but lie outside error bars. This indicates that the homogeneity assumption, while highly accurate, is an excellent approximation yielding numerical values with an error within one part in 10 $^{5}$. Our computed values of the conformal anomaly c and the magnetic scaling dimension X$_{h}$ coincide with the theoretical universality predictions within error bars. q v$_{c}$ (H) v$_{c}$ (N) c (T) c (N) X$_{h}$ (T) X$_{h}$ (N) 0 . 5 0.787417375457453 0.787320(1) -0.445833945 -0.445834(1) 0.082757037 0.082757(1) 1 . 0 1.102738621067509 1.10262924(2) 0 0 0.104166667 0.104167(1) 1 . 5 1.342082948593078 1.3420126(2) 0.288024142 0.2880243(3) 0.116778423 0.116780(3) 2 . 0 1.542459756837412 1.5424598(1) 0.5 0.500000(1) 1/8 0.12500000(1) 2 . 5 1.718102046569530 1.718191(3) 0.66584083 0.66584(1) 0.130338138 0.1304(1) 3 . 0 1.876269208345760 1.876458(3) 4/5 0.8000(1) 2/15 0.1333(1) 3 . 5 2.021253955272383 2.02154(2) 0.910294591 0.910(1) 0.133771753 0.134(1) 4 . 0 2.155842236513638 2.15620(5) 1 1.00(1) 1/8 0.13(1) q v$_{c}$ (H) v$_{c}$ (N) c (T) c (N) X$_{h}$ (T) X$_{h}$ (N) 0 . 5 2.007916417382387 2.00788(1) -0.445833945 -0.4458340(1) 0.082757037 0.08276(1) 1 . 0 2.852426157798754 2.8523883(2) 0 0 0.104166667 0.104167(1) 1 . 5 3.510849695265078 3.510825(2) 0.288024142 0.288024(1) 0.116778423 0.116778(1) 2 . 0 4.073446135573680 4.0734460(1) 0.5 0.500000(1) 1/8 0.12500000(1) 2 . 5 4.574927577671523 4.574952(3) 0.66584083 0.66585(1) 0.130338138 0.13033(1) 3 . 0 5.033022514872745 5.033077(3) 4/5 0.800(1) 2/15 0.13333(1) 3 . 5 5.458234413883058 5.458313(2) 0.910294591 0.91(1) 0.133771753 0.1339(3) 4 . 0 5.857394827983647 5.857497(3) 1 0.999(1) 1/8 0.13(1) 16