Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C8 · Schur multiplier (H2) = C32 |
| direct product | C8×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C32 |
| commutator | C3 | C33×C8 |
| Frattini, cosocle | C3×C4 | C2×C33 |
| socle | C2×C3 | C33×C4 |
| prime radical | C2×H(3,𝔽3) | C3×C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8)(9,10,13)(11,15,21)(12,16,20)(14,22,19)(17,24,29)(18,25,28)(23,30,27)(26,32,34)(31,35,33);(10,14,16)(13,20,22)(15,23,25)(21,28,30)(24,31,32)(29,34,35);(9,11,17,12,18,26,19,27,33)(10,15,24,16,25,32,14,23,31)(13,21,29,20,28,34,22,30,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 250);