Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C52×Q8 ⋊ C22 |
| Sylow | C52 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #87886 |
| commutator, cosocle | C4×C52 | C23 |
| Frattini | C4 | C2×D52 |
| Fitting | C52×Q8 | C22 |
| socle | C2×C52 | C2×D4 |
| hypercenter | Q8 | D52 |
| hypocenter | C52 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
176 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 38 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|c^2=e,d^2=e,c^a=cd,d^a=de,g^a=g^4,c^b=cde,d^b=de,f^b=f^4,d^c=de59 conjugacy classes — view the full character table.
| maximal subgroup of | #13104338, #13104341, #13104343, … |
| maximal normal subgroup of | #13104338, #13104341, #13104343, … |
| maximal quotient of | #13100599, #13100617, #13100647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 718);