Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C5×#400 ⋊ C2 |
| Sylow | C5 ⋊ #2744 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#400 | C2 |
| prime radical | C23×D5 | C23 |
| hypercenter | #400 | D5 |
| hypocenter | C5 | #2744 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #400 | |
392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=fg,c^2=ef,b^a=bef,c^a=cfg,d^a=deg,h^a=h^4,c^b=cg,d^b=def,h^b=h^4,h^d=h^476 conjugacy classes — view the full character table.
| maximal subgroup of | #12306423, #12306436, #12306517, … |
| maximal normal subgroup of | #12306423, #12306436, #12306517, … |
| maximal quotient of | #12026719, #12026767, #12026923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12198);