Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4185 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | #732 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4185 |
| commutator, hypocenter | #732 | C8 |
| Frattini | C32×C4 | C3⋊D3 |
| Fitting, cosocle | C4×#732 | C2 |
| socle | C2×C3 | #1084 |
| prime radical | C2×#732 | C4 |
| LCS | G ▷ #732 | |
| UCS | C1 ◁ C4 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^a=d^2,e^a=e^2,g^a=g^2,e^d=ef,f^d=fg,f^e=fg^252 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊C8, #4179⋊C8, #13046447, … |
| maximal normal subgroup of | #4178⋊C8, #4179⋊C8, #13046447, … |
| maximal quotient of | #13046328, C2×#10619511, #14515658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 30);