Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C8×C27⋊C3 |
| unambiguous semidirect product | (C8×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C32 |
| commutator | C3 | C3×C8×C9 |
| Frattini, cosocle | C4×C9 | C2×C32 |
| socle | C2×C3 | C3×C4×C9 |
| prime radical | C2×C32 | C4×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8)(9,10,13,11,15,21,17,24,29,12,16,20,18,25,28,26,32,34,19,14,22,27,23,30,33,31,35);(10,14,16)(13,20,22)(15,23,25)(21,28,30)(24,31,32)(29,34,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 36);