Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C25 |
| Frattini | C4 | C24×D5 |
| Fitting | C2×C5×#135 | C2 |
| socle | C2×C5 | C23×D4 |
| hypercenter | C2×#135 | D5 |
| hypocenter | C5 | #3584 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#135 | |
1424 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|e^2=g,f^2=g,b^a=bf,f^a=fg,f^b=fg,d^c=dg,h^c=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #12015004, #12015006, #12015009, … |
| maximal normal subgroup of | #12015004, #12015006, #12015009, … |
| maximal quotient of | #12962861, #12962975, #12963414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21170);