Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#478 ⋊ C2 |
| Sylow | C5 ⋊ #3297 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#478 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #478 | D5 |
| hypocenter | C5 | #3297 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #478 | |
420 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,d^b=deg,h^b=h^4,f^c=fg,h^d=h^473 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3297, C2×#10612918, #12975787, … |
| maximal normal subgroup of | D5×#3297, C2×#10612918, #12975787, … |
| maximal quotient of | #12114714, #12115558, #12116483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15000);