Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C9 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63385 |
| commutator, cosocle | C9×C16 | C22 |
| Frattini | C3×C16 | C2×D3 |
| Fitting | C9×Q32 | C2 |
| socle, prime radical | C2×C3 | #826 |
| hypercenter | Q32 | D9 |
| hypocenter | C9 | Q64 |
| LCS | G ▷ C9×C16 ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| 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,g^3=h^2,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2h,h^a=h^2,c^b=cd,d^b=de,e^b=ef63 conjugacy classes — view the full character table.
| maximal subgroup of | D9×Q64, C2×#10588199, #11800553, … |
| maximal normal subgroup of | D9×Q64, C2×#10588199, #11800553, … |
| maximal quotient of | #11772923, #11772926, #11773033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 81);