Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D8×C27 ⋊ C2 |
| Sylow | C27 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D27 |
| commutator, cosocle | C4×C27 | C23 |
| Frattini | C4×C9 | C22×D3 |
| Fitting | D8×C27 | C2 |
| socle | C2×C3 | D4×D9 |
| hypercenter | D8 | D27 |
| hypocenter | C27 | #135 |
| LCS | G ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|d^2=e,f^3=g,g^3=h,b^a=be,c^a=ce,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=cd,d^b=de,d^c=de105 conjugacy classes — view the full character table.
| maximal subgroup of | #13134416, #13134417, #13134420, … |
| maximal normal subgroup of | #13134416, #13134417, #13134420, … |
| maximal quotient of | #13134270, #13134271, #13134272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 128);