Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D33 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×C4 | C23 |
| hypercenter | C4 | D33 |
| hypocenter | C33 | H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
814 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|c^2=e,d^2=e,d^a=de,h^a=h^2,c^b=ce,g^b=g^2,d^c=de,f^c=f^269 conjugacy classes — view the full character table.
| maximal subgroup of | #13167827, #13167830, #13167833, … |
| maximal normal subgroup of | #13167827, #13167830, #13167833, … |
| maximal quotient of | #13167989, #13167995, #13168010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4352);