Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C72 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#87640 |
| commutator, cosocle | C4×C72 | C23 |
| Frattini | C4 | C2×D72 |
| Fitting | C2×C4×C72 | C22 |
| socle | C2×C72 | C2×D4 |
| hypercenter | C2×C4 | D72 |
| hypocenter | C72 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
166 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,7^2|d^2=e,b^a=bd,d^a=de,f^a=f^6,c^b=ce,d^b=de,g^b=g^6143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 455);