Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C32×#127 ⋊ C2 |
| Sylow | C32 ⋊ #540 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#127 | C2 |
| hypercenter | #127 | C3⋊D3 |
| hypocenter | C32 | #540 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #127 | |
720 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 183 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|b^2=f,c^2=e,c^a=cf,d^a=de,g^a=g^2,h^a=h^2,d^b=df,d^c=de78 conjugacy classes — view the full character table.
| maximal subgroup of | #11872246, #11872252, #11872342, … |
| maximal normal subgroup of | #11872246, #11872252, #11872342, … |
| maximal quotient of | #11802840, #11803101, #11803563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8072);