Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C5×#386 ⋊ C2 |
| Sylow | C5 ⋊ #2696 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#386 | C2 |
| prime radical | #65512 | C2 |
| hypercenter | #386 | D5 |
| hypocenter | C5 | #2696 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #386 | |
562 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=ef,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=def,h^a=h^4,c^b=cg,d^b=de,h^c=h^4,h^d=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #12306134, #12306142, #12306156, … |
| maximal normal subgroup of | #12306134, #12306142, #12306156, … |
| maximal quotient of | #12026619, #12026792, #12026798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 8622);