Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #41×C27 ⋊ C2 |
| Sylow | C27 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#6171 |
| commutator, cosocle | C4×C27 | C23 |
| Frattini | C4×C9 | C22×D3 |
| Fitting | #41×C27 | C2 |
| socle | C2×C3 | C2×#611 |
| hypercenter | #41 | D27 |
| hypocenter | C27 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
136 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^5,3^3|d^2=e,f^3=g,g^3=h,c^a=cd,d^a=de,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=ce,d^c=de141 conjugacy classes — view the full character table.
| maximal subgroup of | #13134327, #13134332, #13134647, … |
| maximal normal subgroup of | #13134327, #13134332, #13134647, … |
| maximal quotient of | #13134502, #13134507, #13134509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 166);