Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C5 |
No aliases for this group.
| Fitting | C52×C4⋊C4 ⋊ C2 |
| Sylow | C52 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C23 |
| Fitting | C52×C4⋊C4 | C2 |
| prime radical | C2×C4⋊(C5⋊D5) | C2 |
| hypercenter | C4⋊C4 | C5⋊D5 |
| hypocenter | C52 | #123 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
312 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^5,5^2|b^2=d,c^2=e,c^a=cd,f^a=f^4,g^a=g^4,c^b=ce134 conjugacy classes — view the full character table.
| maximal subgroup of | #13105425, #13105429, #13105471, … |
| maximal normal subgroup of | #13105425, #13105429, #13105471, … |
| maximal quotient of | #13102527, #13102529, #13102531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 873);