Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C52 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #87771 |
| commutator, Fitting, cosocle | C52×C8 | C22 |
| Frattini, hypercenter | C8 | D52 |
| socle, prime radical | C2×C52 | D8 |
| hypocenter | C52 | Q32 |
| LCS | G ▷ C52×C8 ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^5,5^2|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,g^a=g^4,f^b=f^489 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic40, #13100419, #13100420, … |
| maximal normal subgroup of | D5×Dic40, #13100419, #13100420, … |
| maximal quotient of | #13099288, #13099289, #13099309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 251);