Finite Group DatabasebetaFinite Group Databasebeta| order | 8144 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C509 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D509 |
| commutator | C2×C509 | C2×C4 |
| Fitting | C2×C4×C509 | C2 |
| socle | C22×C509 | C22 |
| hypercenter | C2×C4 | D509 |
| hypocenter | C509 | C4⋊C4 |
| LCS | G ▷ C2×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
26 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,509|a^2=d,b^2=c,b^a=bc,e^b=e^5082042 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4⋊C4×D509, C2×#22550840, #30654171, … |
| maximal normal subgroup of | C4⋊C4×D509, C2×#22550840, #30654171, … |
| maximal quotient of | #30654122, #30654144, #30654146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8144, 19);