Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C33 |
No aliases for this group.
| Fitting | C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×#41 | C2 |
| hypercenter | #41 | C3⋊(C3⋊D3) |
| hypocenter | C33 | H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ #41 | |
2324 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 527 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,d^a=de,f^a=f^2,g^a=g^2,h^a=h^2,c^b=ce147 conjugacy classes — view the full character table.
| maximal subgroup of | #13161357, #13161358, #13161404, … |
| maximal normal subgroup of | #13161357, #13161358, #13161404, … |
| maximal quotient of | #13179644, #13179647, #13179649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4651);