Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×#508 ⋊ C2 |
| Sylow | C9 ⋊ #2263 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#63385 |
| commutator, cosocle | C9×C16 | C23 |
| Frattini | C3×C16 | C22×D3 |
| Fitting | C9×#508 | C2 |
| socle | C2×C3 | C2×#826 |
| hypercenter | #508 | D9 |
| hypocenter | C9 | #2263 |
| LCS | G ▷ C9×C16 ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #508 | |
168 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 56 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:
2^7,3^2|d^2=ef,e^2=fg,f^2=g,h^3=i,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg,h^b=h^2i^2,i^b=i^2114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 34594);