Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C32 ⋊ #340 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | AΓL(1,𝔽9) |
| commutator | #157 | C42 |
| Fitting, socle | C22×C32 | SD16 |
| prime radical | C22×C3⋊D3 | D4 |
| cosocle | C22×#157 | C22 |
| hypocenter | C32 | #340 |
| LCS | G ▷ #157 ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22 | |
141 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^6,3^2|a^2=c,b^2=d,c^2=f,b^a=bcef,g^a=g^2h^2,h^a=g^2,c^b=cf,g^b=gh^2,h^b=h^2,g^c=g^2h,h^c=gh,g^f=g^2,h^f=h^236 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10593166, C4×GO+(4,𝔽2)⋊C4, C4×PSU(3,𝔽2)⋊C4, … |
| maximal normal subgroup of | C2×#10593166, C4×GO+(4,𝔽2)⋊C4, C4×PSU(3,𝔽2)⋊C4, … |
| maximal quotient of | #11770760, #11770801, #11770802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5048);