Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C5×#376 ⋊ C2 |
| Sylow | C5 ⋊ #2397 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Frattini | C23 | C23×D5 |
| Fitting | C5×#376 | C2 |
| hypercenter | #376 | D5 |
| hypocenter | C5 | #2397 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
840 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^4,h^c=h^4124 conjugacy classes — view the full character table.
| maximal subgroup of | #11994378, #12000140, #12000143, … |
| maximal normal subgroup of | #11994378, #12000140, #12000143, … |
| maximal quotient of | #12021036, #12021082, C4×#64960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 7231);