Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #64230 |
| commutator, cosocle | C4×C3⋊D3 | C23 |
| Frattini | C4 | C2×GO+(4,𝔽2) |
| Fitting | C32×C8 | D4 |
| socle | C2×C32 | #121 |
| hypercenter | C8 | GO+(4,𝔽2) |
| hypocenter | C32 | #472 |
| LCS | G ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
177 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|c^2=ef,d^2=f,e^2=f,b^a=bd,c^a=cf,d^a=df,g^a=g^2,c^b=ce,d^b=df,e^b=ef,g^b=h,h^b=g,g^d=g^2,h^d=h^233 conjugacy classes — view the full character table.
| maximal subgroup of | #11798370, #11798417, C2×#10593303, … |
| maximal normal subgroup of | #11798370, #11798417, C2×#10593303, … |
| maximal quotient of | #11780902, #11780934, #11781192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5185);