Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C32 |
| direct product | C32×#622 |
| derived | C32 ⋊ C32×C8 |
| Sylow | C34 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | #157 |
| commutator, hypocenter | C32 | C32×C8 |
| Frattini | C2 | C32×#157 |
| Fitting, socle, prime radical | C2×C34 | C4 |
| cosocle | C3⋊Dic3 | C2×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C32 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#3771, C3×C3⋊#3771, C3⋊(C3⋊#3771), … |
| maximal normal subgroup of | C32×#3771, C3×C3⋊#3771, C3⋊(C3⋊#3771), … |
| maximal quotient of | C32×#3708, C2×C32×#622, C3×C9×#622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 642);