Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#1113 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×Q8 | C22 |
| hypercenter | Q8 | #1113 |
| hypocenter | C33 | 2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
712 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^2,h^a=h^2,c^b=ce,f^b=f^2,g^b=g^2,d^c=de72 conjugacy classes — view the full character table.
| maximal subgroup of | #13167266, #13167268, #13167559, … |
| maximal normal subgroup of | #13167266, #13167268, #13167559, … |
| maximal quotient of | #13178296, #13178297, #13178299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4591);