Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C9×Q64 ⋊ C2 |
| Sylow | C9 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10588196 |
| commutator, cosocle | C9×C32 | C22 |
| Frattini | C3×C32 | C2×D3 |
| Fitting | C9×Q64 | C2 |
| socle | C2×C3 | #4519 |
| prime radical | D288 | C2 |
| hypercenter | Q64 | D9 |
| hypocenter | C9 | SD128 |
| LCS | G ▷ C9×C32 ▷ C9×C16 ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^7,3^2|a^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,h^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2i^2,i^b=i^2111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 7126);