Finite Group DatabasebetaFinite Group Databasebeta| order | 8144 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D509⋊C8 |
| derived, Sylow | C509 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #14523266 |
| commutator, hypocenter | C509 | C2×C8 |
| Frattini | C2 | C2×#14523266 |
| Fitting | C4×C509 | C4 |
| socle | C2×C509 | C2×C4 |
| LCS | G ▷ C509 | |
| UCS | C1 ◁ C4 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,509|a^2=c,c^2=d,e^a=e^301,e^b=e^508,e^c=e^508524 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#14523266, #30654269, #30654277, … |
| maximal normal subgroup of | C8×#14523266, #30654269, #30654277, … |
| maximal quotient of | C4×#16861332, #30654272, #30654282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8144, 48);