Finite Group DatabasebetaFinite Group Databasebeta| order | 8656 |
| automorphism | Aut(G) order 9348480 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4328 |
| Fitting | C8×C541 ⋊ C2 |
| Sylow | C541 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2164 |
| commutator | C4×C541 | C22 |
| Frattini | C4 | C2×D541 |
| Fitting | C8×C541 | C2 |
| socle | C2×C541 | D4 |
| hypercenter | C8 | D541 |
| hypocenter | C541 | D8 |
| LCS | G ▷ C4×C541 ▷ C2×C541 ▷ C541 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,541|c^2=d,b^a=bc,c^a=cd,e^a=e^540,c^b=cd,e^b=e^5402167 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #31772569, D8656, #31772571, … |
| maximal normal subgroup of | #31772569, D8656, #31772571, … |
| maximal quotient of | #31772550, #31772562, D8656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8656, 26);