Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C9×C22⋊Q8 ⋊ C2 |
| Sylow | C9 ⋊ #552 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C9×C22⋊Q8 | C2 |
| prime radical | C2×#611 | C22 |
| hypercenter | C22⋊Q8 | D9 |
| hypocenter | C9 | #552 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
321 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^2|c^2=e,d^2=ef,g^3=h,b^a=bf,d^a=de,g^a=g^2h^2,h^a=h^2,d^b=df,d^c=de81 conjugacy classes — view the full character table.
| maximal subgroup of | #11887209, #11887215, #11887720, … |
| maximal normal subgroup of | #11887209, #11887215, #11887720, … |
| maximal quotient of | #11805055, #11805160, #11805285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1626);