Finite Group DatabasebetaFinite Group Databasebeta| order | 8144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | Q8×C509 ⋊ C2 |
| Sylow | C509 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D509 |
| commutator | C4×C509 | C22 |
| Frattini | C4 | C2×D509 |
| Fitting | Q8×C509 | C2 |
| socle | C2×C509 | D4 |
| hypercenter | Q8 | D509 |
| hypocenter | C509 | SD16 |
| LCS | G ▷ C4×C509 ▷ C2×C509 ▷ C509 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
20 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,509|a^2=d,c^2=d,b^a=bc,c^a=cd,c^b=cd,e^b=e^5081277 conjugacy classes. The character table is too large to display.
| maximal subgroup of | SD16×D509, C2×#22550848, #30654235, … |
| maximal normal subgroup of | SD16×D509, C2×#22550848, #30654235, … |
| maximal quotient of | #30654135, #30654139, #30654146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8144, 27);