Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D8×C31 ⋊ C2 |
| Sylow | C31 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D31 |
| commutator, cosocle | C4×C31 | C23 |
| Frattini | C4 | C22×D31 |
| Fitting | D8×C31 | C2 |
| socle | C2×C31 | C2×D4 |
| hypercenter | D8 | D31 |
| hypocenter | C31 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C31 ▷ C2×C31 ▷ C31 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 248 |
Polycyclic presentation:
2^5,31|d^2=e,c^a=ce,f^a=f^30,c^b=cd,d^b=de,d^c=de116 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×D31, C2×#11758488, #14521845, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×D31, C2×#11758488, #14521845, … |
| maximal quotient of | #14521047, #14521052, #14521090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 106);