Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#624 |
| derived | C32 ⋊ C2×C32×C4 |
| Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D32 |
| commutator, hypocenter | C32 | C2×C32×C4 |
| Frattini | C2 | C32×D32 |
| Fitting, socle | C2×C34 | C22 |
| prime radical | C2×C32×C3⋊D3 | C2 |
| cosocle | C2×C32 | C22×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C32 | |
374 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
162 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D32⋊C4, C32×#3773, C3×#88932, … |
| maximal normal subgroup of | C32×D32⋊C4, C32×#3773, C3×#88932, … |
| maximal quotient of | C32×#3710, C32×#3712, C32×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 647);