Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C4×C16 |
| Frattini | C4 | C2×AGL(1,𝔽9) |
| Fitting | C32×C8 | C8 |
| socle | C2×C32 | C4×C8 |
| prime radical | C2×C3⋊D3 | C2×C8 |
| cosocle | #3787 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
69 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|a^2=c,b^2=d,c^2=e,d^2=f,e^2=f,g^a=g^2h^2,h^a=g^2,g^c=g^2h,h^c=gh,g^e=g^2,h^e=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | C16×AGL(1,𝔽9), #11771303, GO+(4,𝔽2)⋊C16, … |
| maximal normal subgroup of | C16×AGL(1,𝔽9), #11771303, GO+(4,𝔽2)⋊C16, … |
| maximal quotient of | C8×#3771, #11771317, #11772269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5136);