Finite Group DatabasebetaFinite Group Databasebeta| order | 5072 |
| automorphism | Aut(G) order 3205504 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D2536 |
| Fitting | C8×C317 ⋊ C2 |
| Sylow | C317 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D1268 |
| commutator | C4×C317 | C22 |
| Frattini | C4 | C2×D317 |
| Fitting | C8×C317 | C2 |
| socle | C2×C317 | D4 |
| hypercenter | C8 | D317 |
| hypocenter | C317 | D8 |
| LCS | G ▷ C4×C317 ▷ C2×C317 ▷ C317 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,317|c^2=d,b^a=bc,c^a=cd,e^a=e^316,c^b=cd,e^b=e^3161271 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D7608, D12680, D17752, … |
| maximal quotient of | D7608, D12680, D17752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5072, 26);