Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C32×#125 ⋊ C2 |
| Sylow | C32 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#125 | C2 |
| prime radical | C2×C4⋊(C3⋊D3) | C22 |
| hypercenter | #125 | C3⋊D3 |
| hypocenter | C32 | #563 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #125 | |
588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 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=e,c^2=ef,d^2=f,c^a=cf,d^a=def,g^a=g^2,h^a=h^2,c^b=ce,d^b=df75 conjugacy classes — view the full character table.
| maximal subgroup of | #11780512, #11780521, #11780529, … |
| maximal normal subgroup of | #11780512, #11780521, #11780529, … |
| maximal quotient of | #11807632, #11807793, #11807799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8052);