Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C5×C4⋊Q8 ⋊ C2 |
| Sylow | C5 ⋊ #3499 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C2×C5×C4⋊Q8 | C2 |
| hypercenter | C2×C4⋊Q8 | D5 |
| hypocenter | C5 | #3499 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊Q8 | |
1476 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #11991216, #11991220, #11991527, … |
| maximal normal subgroup of | #11991216, #11991220, #11991527, … |
| maximal quotient of | #12259402, #12259669, #12259673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20197);