Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C27 · Schur multiplier (H2) = C3 |
| direct product | C3×Q8×C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C27 | C22 |
| commutator | C2 | C22×C3×C27 |
| Frattini, cosocle | C2×C9 | C22×C32 |
| socle, prime radical | C2×C32 | C22×C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C27 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3)(31,32,33,34)(35,38,37,36);(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,35,33,37)(32,36,34,38)405 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 226);