Finite Group DatabasebetaFinite Group Databasebeta| order | 5056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Q8⋊(C79⋊C8) |
| Sylow | C79 ⋊ Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊D79 |
| commutator | C4×C79 | C2×C8 |
| Frattini | C42 | C2×D79 |
| Fitting | C4×Q8×C79 | C2 |
| socle | C22×C79 | C22⋊C4 |
| hypercenter | C4×Q8 | D79 |
| hypocenter | C79 | Q8⋊C8 |
| LCS | G ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×Q8 | |
70 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,79|a^2=d,b^2=e,c^2=e,d^2=f,b^a=bc,c^a=ce,g^a=g^78,c^b=ce808 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5056, 280);