Compact Braille lookup
Just turning some of the comments into an answer, more specific I'm going to address these issues in your code:


No need to double the lookup table – By using char.ToLower(c) you could half the size of the lookup table.
Shift the offset by (char) 'a' – If you shift the offset by the value of a you get a zero-based index.
No need longer for the char dictionary – Since we now have a zero-based index we could simply do a lookup in a byte table. (Update: And it could be a simple array also, no need for a List&lt;byte&gt;)
Since we shifted the offset, no need to include empty entries – By shifting the offset, we don't need a sparse 256 byte array. It does however, fail miserably like your code if trying to convert non-alpha characters.
Wouldn't it be nice to convert longer texts also? – Wouldn't you need to convert longer texts, and not only single characters? For good measure, I've added a ToString(string) to the code below.


This leads to the following code (extending on the char value):

using System.Text;

public static class MyExtensions
{
    static byte[] braille = new byte[]
      { 0x20, 0x30, 0x24, 0x26, 0x22, 0x34, 0x36, 0x32, 0x14, 0x16,
        0x28, 0x38, 0x2c, 0x2e, 0x2a, 0x3c, 0x3e, 0x3a, 0x1c, 0x1e,
        0x29, 0x39, 0x17, 0x2d, 0x2f, 0x2b
      };        

    public static string ToBraille(this char c) {
      return Convert.ToString(braille[ (byte) char.ToLower(c) - (byte) 'a'], 2).PadLeft(6, '0');
    }

    public static string ToBraille(this string s) {
      StringBuilder result = new StringBuilder();
      foreach (char c in s) {
        result.Append(c.ToBraille()).Append(" ");
      }

      return result.ToString();
    }
}