Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×AGL(1,𝔽9) |
| commutator, socle | C2×C32 | C22×C8 |
| Fitting | C32×D4 | C8 |
| prime radical | #3830 | C4 |
| cosocle | #622 | C23 |
| hypercenter | D4 | AGL(1,𝔽9) |
| hypocenter | C32 | #504 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|a^2=d,d^2=e,e^2=f,g^a=g^2h^2,h^a=g^2,c^b=cf,g^d=g^2h,h^d=gh,g^e=g^2,h^e=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | #11797577, #11797578, #11797613, … |
| maximal normal subgroup of | #11797577, #11797578, #11797613, … |
| maximal quotient of | #11795905, #11795906, #11796016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8367);