Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C5 |
No aliases for this group.
| Fitting | C52×D8 ⋊ C2 |
| Sylow | C52 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C5⋊D5 |
| commutator, cosocle | C4×C52 | C23 |
| Frattini | C4 | C22×C5⋊D5 |
| Fitting | C52×D8 | C2 |
| socle | C2×C52 | C2×D4 |
| hypercenter | D8 | C5⋊D5 |
| hypocenter | C52 | #135 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
248 conjugacy classes of subgroups.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^5,5^2|d^2=e,b^a=be,c^a=ce,f^a=f^4,g^a=g^4,c^b=cd,d^b=de,d^c=de98 conjugacy classes — view the full character table.
| maximal subgroup of | #13100281, #13100283, #13100373, … |
| maximal normal subgroup of | #13100281, #13100283, #13100373, … |
| maximal quotient of | #13102628, #13102629, #13102630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 891);