Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C32×C22⋊Q8 ⋊ C2 |
| Sylow | C32 ⋊ #543 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×C22⋊Q8 | C2 |
| prime radical | C23×C3⋊D3 | C22 |
| hypercenter | C22⋊Q8 | C3⋊D3 |
| hypocenter | C32 | #543 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
726 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 191 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|c^2=f,d^2=f,d^a=de,g^a=g^2,h^a=h^2,c^b=ce,d^c=df78 conjugacy classes — view the full character table.
| maximal subgroup of | #11778009, #11778018, #11778026, … |
| maximal normal subgroup of | #11778009, #11778018, #11778026, … |
| maximal quotient of | #11802885, #11802891, #11803885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7987);