Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52⋊C2 |
| commutator, cosocle | #1039 | C23 |
| Fitting | C22×C52 | D4 |
| socle | C2×C52 | C2×D4 |
| hypercenter | C22 | D52⋊C2 |
| hypocenter | C52 | AGL(1,ℤ/8) |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
125 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 23 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|d^2=e,b^a=bd,d^a=de,f^a=f^4,g^a=fg,c^b=ce,d^b=de,f^b=f^2g^4,g^b=f^3g^3,f^d=f^4,g^d=g^441 conjugacy classes — view the full character table.
| maximal subgroup of | #13103391, #13103392, #13104314, … |
| maximal normal subgroup of | #13103391, #13103392, #13104314, … |
| maximal quotient of | #13104189, #13104191, #13104207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1053);