Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C4×C52 ⋊ D4 |
| Sylow | C52 ⋊ #127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C23 |
| Fitting | C4×C52 | D4 |
| socle | C2×C52 | C2×D4 |
| hypercenter | C4 | D52⋊C2 |
| hypocenter | C52 | #127 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
177 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^5,5^2|c^2=e,b^a=bd,c^a=ce,f^a=f^4,g^a=fg,c^b=ce,f^b=f^2g^4,g^b=f^3g^3,f^d=f^4,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #13103320, #13103321, #13103346, … |
| maximal normal subgroup of | #13103320, #13103321, #13103346, … |
| maximal quotient of | #13104153, #13104154, #13104155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1048);