Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C5×C32 ⋊ C2 |
| Sylow | C5 ⋊ #2262 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D80 |
| commutator, cosocle | C5×C16 | C23 |
| Frattini | C16 | C22×D5 |
| Fitting | C2×C5×C32 | C2 |
| socle | C2×C5 | C2×D16 |
| hypercenter | C2×C32 | D5 |
| hypocenter | C5 | #2262 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,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^4166 conjugacy classes — view the full character table.
| maximal subgroup of | #11987183, #11987185, #11987186, … |
| maximal normal subgroup of | #11987183, #11987185, #11987186, … |
| maximal quotient of | #12019735, #12019803, C4×D160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2244);