Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #66343 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C9 |
No aliases for this group.
| derived, Sylow | C92 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #66343 |
| commutator, hypocenter | C92 | C8 |
| Frattini, socle, prime radical | C2×C32 | #157 |
| Fitting | C2×C92 | C4 |
| cosocle | C9⋊Dic9 | C2 |
| LCS | G ▷ C92 | |
| UCS | C1 ◁ C2 | |
42 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^3=f,e^3=g,d^a=deg^2,e^a=de^2g^2,f^a=fg,g^a=fg^2,d^b=d^2f^2,e^b=e^2g^2,f^b=f^2,g^b=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #13046936, #13046939, #13046940, … |
| maximal normal subgroup of | #13046936, #13046939, #13046940, … |
| maximal quotient of | #13046359, C2×#10619526, #14515710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 45);