Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52⋊C2 |
| commutator, cosocle | #1039 | C23 |
| Fitting | C22×C52 | D4 |
| socle | C2×C52 | C2×D4 |
| hypercenter | C22 | D52⋊C2 |
| hypocenter | C52 | #135 |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
113 conjugacy classes of subgroups.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^5,5^2|d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^4,g^a=fg,c^b=ce,d^b=de,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 | #13103389, #13103390, #13107664, … |
| maximal normal subgroup of | #13103389, #13103390, #13107664, … |
| maximal quotient of | #13104190, #13104195, #13104231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1054);