Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C5 |
| unambiguous semidirect product | C4⋊#6057 |
| Fitting | C4×C52 ⋊ Q8 |
| Sylow | C52 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#6057 |
| commutator, prime radical, cosocle | C2×C5⋊D5 | C23 |
| Fitting | C4×C52 | Q8 |
| socle | C2×C52 | C2×Q8 |
| hypercenter | C4 | #6057 |
| hypocenter | C52 | C4⋊Q8 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
109 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^5,5^2|a^2=d,b^2=d,c^2=e,b^a=bd,f^a=f^2,g^a=g^3,c^b=ce,f^b=g,g^b=f^4,f^d=f^4,g^d=g^426 conjugacy classes — view the full character table.
| maximal subgroup of | #13103332, #13103333, #13103334, … |
| maximal normal subgroup of | #13103332, #13103333, #13103334, … |
| maximal quotient of | #13104396, #13104397, #13104398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1062);