Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×M64 ⋊ C2 |
| Sylow | C5 ⋊ #2263 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D80 |
| commutator, cosocle | C5×C16 | C23 |
| Frattini | C16 | C22×D5 |
| Fitting | C5×M64 | C2 |
| socle | C2×C5 | C2×D16 |
| hypercenter | M64 | D5 |
| hypocenter | C5 | #2263 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ M64 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^4,d^b=de,e^b=ef,f^b=fg,h^b=h^4106 conjugacy classes — view the full character table.
| maximal subgroup of | #12019924, #12019927, #12019974, … |
| maximal normal subgroup of | #12019924, #12019927, #12019974, … |
| maximal quotient of | #12019740, #12019744, #12019807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2248);