Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3 | PSU(3,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | SU(3,𝔽2) | C3 |
| Fitting | H(3,𝔽3) | SL(2,𝔽3) |
| prime radical | #272 | A4 |
| LCS | G ▷ SU(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3,2^3,3^3|a^3=g^2,b^2=d,c^2=d,b^a=c,c^a=bcd,e^a=fg^2,f^a=e^2f^2g,c^b=cd,e^b=ef,f^b=ef^2g,e^c=fg^2,f^c=e^2,e^d=e^2g^2,f^d=f^2g,f^e=fg24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 531);