Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D5×AGL(1,𝔽5) |
| commutator | C2×C52 | C2×C8 |
| Fitting, socle | C22×C52 | C2×C4 |
| prime radical | C22×C5⋊D5 | C4 |
| cosocle | C2×C5×Dic5 | C22 |
| hypocenter | C52 | C22⋊C8 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 | |
116 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^5,5^2|a^2=c,b^2=e,c^2=e,b^a=bd,g^a=g^2,f^b=f^4,g^c=g^468 conjugacy classes — view the full character table.
| maximal subgroup of | #13103482, #13103484, #13103494, … |
| maximal normal subgroup of | #13103482, #13103484, #13103494, … |
| maximal quotient of | #13099752, #13099753, #13099756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 413);