Finite Group DatabasebetaFinite Group Databasebeta| order | 7856 |
| automorphism | Aut(G) order 7698880 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3928 |
| Fitting | C8×C491 ⋊ C2 |
| Sylow | C491 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D1964 |
| commutator | C4×C491 | C22 |
| Frattini | C4 | C2×D491 |
| Fitting | C8×C491 | C2 |
| socle | C2×C491 | D4 |
| hypercenter | C8 | D491 |
| hypocenter | C491 | D8 |
| LCS | G ▷ C4×C491 ▷ C2×C491 ▷ C491 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,491|c^2=d,b^a=bc,c^a=cd,e^a=e^490,c^b=cd,e^b=e^4901967 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30628278, D7856, #30628280, … |
| maximal normal subgroup of | #30628278, D7856, #30628280, … |
| maximal quotient of | #30628259, #30628271, D7856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7856, 26);