Finite Group DatabasebetaFinite Group Databasebeta| order | 656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | D4×C41 ⋊ C2 |
| Sylow | C41 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D41 |
| commutator, cosocle | C4×C41 | C22 |
| Frattini | C4 | C2×D41 |
| Fitting | D4×C41 | C2 |
| socle | C2×C41 | D4 |
| hypercenter | D4 | D41 |
| hypocenter | C41 | D8 |
| LCS | G ▷ C4×C41 ▷ C2×C41 ▷ C41 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 328 |
Polycyclic presentation:
2^4,41|c^2=d,b^a=bc,c^a=cd,e^a=e^40,c^b=cd107 conjugacy classes — view the full character table.
| maximal subgroup of | D8×D41, C2×#10620282, #13050184, … |
| maximal normal subgroup of | D8×D41, C2×#10620282, #13050184, … |
| maximal quotient of | #13050084, #13050085, #13050097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(656, 15);