Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C32×C8 ⋊ D4 |
| Sylow | C32 ⋊ #493 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #64231 |
| commutator, cosocle | C4×C3⋊D3 | C23 |
| Frattini | C4 | C2×GO+(4,𝔽2) |
| Fitting | C32×C8 | D4 |
| socle | C2×C32 | #127 |
| hypercenter | C8 | GO+(4,𝔽2) |
| hypocenter | C32 | #493 |
| LCS | G ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
238 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|c^2=ef,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,g^b=h,h^b=g,g^d=g^2,h^d=h^236 conjugacy classes — view the full character table.
| maximal subgroup of | #11772015, #11772072, #11780937, … |
| maximal normal subgroup of | #11772015, #11772072, #11780937, … |
| maximal quotient of | #11782459, #11782461, #11782547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5175);