Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63653 |
| commutator, cosocle | C32×C16 | C22 |
| Frattini | C16 | C2×C3⋊D3 |
| Fitting | C32×Q32 | C2 |
| socle, prime radical | C2×C32 | D16 |
| hypercenter | Q32 | C3⋊D3 |
| hypocenter | C32 | Q64 |
| LCS | G ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^2,c^b=cd,d^b=de,e^b=ef63 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×Q64, C2×#10589388, D3×#4522, … |
| maximal normal subgroup of | C3⋊D3×Q64, C2×#10589388, D3×#4522, … |
| maximal quotient of | #11772924, #11772927, #11773034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1270);