Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C5 |
No aliases for this group.
| Sylow | C52 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#1044 |
| commutator, socle | C2×C52 | C22×C4 |
| Fitting | C52×Q8 | C4 |
| prime radical | #87904 | C2 |
| cosocle | #1039 | C23 |
| hypercenter | Q8 | #1044 |
| hypocenter | C52 | #131 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
164 conjugacy classes of subgroups.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^5,5^2|a^2=d,b^2=e,c^2=e,d^2=e,f^a=f^3,g^a=g^2,c^b=ce,f^d=f^4,g^d=g^450 conjugacy classes — view the full character table.
| maximal subgroup of | #13104085, #13104086, #13104341, … |
| maximal normal subgroup of | #13104085, #13104086, #13104341, … |
| maximal quotient of | #13104839, #13104840, #13104841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1113);