Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C5×#392 ⋊ C2 |
| Sylow | C5 ⋊ #2699 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#392 | C2 |
| hypercenter | #392 | D5 |
| hypocenter | C5 | #2699 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #392 | |
612 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=f,c^2=ef,b^a=bef,c^a=cf,d^a=de,h^a=h^4,c^b=cg,d^b=def,h^b=h^4,h^c=h^476 conjugacy classes — view the full character table.
| maximal subgroup of | #12306229, #12306233, #12306235, … |
| maximal normal subgroup of | #12306229, #12306233, #12306235, … |
| maximal quotient of | #12026665, #12026668, #12026720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 10533);