Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C23 |
| Fitting | C22×C52 | D4 |
| socle | C2×C52 | C2×D4 |
| prime radical | #87886 | C2 |
| hypercenter | C22 | D52⋊C2 |
| hypocenter | C52 | #123 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
149 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^5,5^2|b^2=e,b^a=bd,c^a=ce,f^a=f^4,g^a=fg,f^b=f^2g^4,g^b=f^3g^3,f^d=f^4,g^d=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #13104273, #13104378, #13107500, … |
| maximal normal subgroup of | #13104273, #13104378, #13107500, … |
| maximal quotient of | #13104171, #13104188, #13104193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1050);