Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C32×#126 ⋊ C2 |
| Sylow | C32 ⋊ #560 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#126 | C2 |
| hypercenter | #126 | C3⋊D3 |
| hypocenter | C32 | #560 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #126 | |
756 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|c^2=f,d^2=ef,c^a=cf,d^a=def,g^a=g^2,h^a=h^2,c^b=ce,d^b=df75 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#560, C2×#10596182, D3×#5711, … |
| maximal normal subgroup of | C3⋊D3×#560, C2×#10596182, D3×#5711, … |
| maximal quotient of | #11807587, #11807632, #11807708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8064);