Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C9×D8 ⋊ C2 |
| Sylow | C9 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63500 |
| commutator, cosocle | C2×C4×C9 | C23 |
| Frattini | C2×C3×C4 | C22×D3 |
| Fitting | C2×C9×D8 | C2 |
| socle | C2×C3 | #938 |
| hypercenter | C2×D8 | D9 |
| hypocenter | C9 | #472 |
| LCS | G ▷ C2×C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
162 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|b^2=e,d^2=f,e^2=f,g^3=h^2,c^a=cd,d^a=df,g^a=g^2h,h^a=h^2,c^b=ce,d^c=df,e^c=ef72 conjugacy classes — view the full character table.
| maximal subgroup of | #11798324, #11798330, #11798371, … |
| maximal normal subgroup of | #11798324, #11798330, #11798371, … |
| maximal quotient of | #11780873, #11780905, #11781163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 722);