Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4183 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | AΣL(1,𝔽27) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4183 |
| commutator, hypocenter | AΣL(1,𝔽27) | C8 |
| Frattini | C32×C4 | C3⋊D3 |
| Fitting, cosocle | C4×AΣL(1,𝔽27) | C2 |
| socle | C2×C3 | #1084 |
| prime radical | C2×AΣL(1,𝔽27) | C4 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C4 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=fg^2,g^a=g^2,e^d=ef,f^d=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C8, #4175⋊C8, #13046411, … |
| maximal normal subgroup of | #4174⋊C8, #4175⋊C8, #13046411, … |
| maximal quotient of | #13046326, C2×#10619509, #14515658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 28);