Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C23×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #3486 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C23×C5×Q8 | C2 |
| hypercenter | C23×Q8 | D5 |
| hypocenter | C5 | #3486 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|b^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^4,d^b=df124 conjugacy classes — view the full character table.
| maximal subgroup of | #11991436, #11991438, #12014842, … |
| maximal normal subgroup of | #11991436, #11991438, #12014842, … |
| maximal quotient of | #12259325, #12259582, #12260238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21234);