Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊((C22⋊C9)⋊C2) |
| commutator, cosocle | #41⋊C9 | C22 |
| Frattini | C3×C4 | C2×S4 |
| Fitting | C3×#131 | D3 |
| socle | C2×C3 | C4⋊S4 |
| hypercenter | C8 | (C22⋊C9)⋊C2 |
| hypocenter | Q8⋊C9 | D4 |
| LCS | G ▷ #41⋊C9 ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
111 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^3,3^2,2^3|b^2=ch,c^2=h,d^3=e^2,f^2=h,g^2=h,b^a=bc,c^a=ch,d^a=d^2e,e^a=e^2,f^a=gh,g^a=fh,f^d=g,g^d=fgh,g^f=gh48 conjugacy classes — view the full character table.
| maximal subgroup of | #11919996, #11919997, #11920093, … |
| maximal normal subgroup of | #11919996, #11919997, #11920093, … |
| maximal quotient of | #11919968, #11919977, #11919978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1391);