Finite Group DatabasebetaFinite Group Databasebeta| order | 656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C41 ⋊ C2 |
| Sylow | C41 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D41 |
| commutator, socle, cosocle | C2×C41 | C23 |
| Fitting | Q8×C41 | C2 |
| hypercenter | Q8 | D41 |
| hypocenter | C41 | #41 |
| LCS | G ▷ C2×C41 ▷ C41 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 328 |
Polycyclic presentation:
2^4,41|b^2=d,c^2=d,c^a=cd,e^a=e^40,c^b=cd110 conjugacy classes — view the full character table.
| maximal subgroup of | #13050185, #13050187, #13050191, … |
| maximal normal subgroup of | #13050185, #13050187, #13050191, … |
| maximal quotient of | #13050122, #13050126, Q8×Dic41, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(656, 43);