Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×(C24⋊C5)⋊C2 |
| commutator, cosocle, hypocenter | 2−1+4⋊C5 | C22 |
| Fitting | #585 | D5 |
| prime radical | #585⋊C5 | C2 |
| hypercenter | C4 | (C24⋊C5)⋊C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
203 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^2,5,2^5|a^2=h,b^2=h,d^2=h,e^2=h,f^2=h,b^a=bh,c^a=c^4,d^a=eh,e^a=dh,f^a=efg,g^a=degh,d^c=e,e^c=ef,f^c=fg,g^c=dgh,e^d=eh,f^d=fh,g^d=gh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #13046157, #13046158, #13046165, … |
| maximal normal subgroup of | #13046157, #13046158, #13046165, … |
| maximal quotient of | #13046126, #13046128, #13046130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21463);