Finite Group DatabasebetaFinite Group Databasebeta| order | 8144 |
| automorphism | Aut(G) order 8274304 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D4072 |
| Fitting | C8×C509 ⋊ C2 |
| Sylow | C509 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2036 |
| commutator | C4×C509 | C22 |
| Frattini | C4 | C2×D509 |
| Fitting | C8×C509 | C2 |
| socle | C2×C509 | D4 |
| hypercenter | C8 | D509 |
| hypocenter | C509 | D8 |
| LCS | G ▷ C4×C509 ▷ C2×C509 ▷ C509 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,509|c^2=d,b^a=bc,c^a=cd,e^a=e^508,c^b=cd,e^b=e^5082039 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30654156, D8144, #30654158, … |
| maximal normal subgroup of | #30654156, D8144, #30654158, … |
| maximal quotient of | #30654137, #30654149, D8144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8144, 26);