Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C52 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D52 |
| commutator, socle, cosocle | C2×C52 | C24 |
| Fitting | C52×D4 | C22 |
| hypercenter | D4 | D52 |
| hypocenter | C52 | 2−1+4 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ D4 | |
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^5,5^2|b^2=e,c^2=e,d^a=de,g^a=g^4,c^b=ce,f^b=f^4,d^c=de77 conjugacy classes — view the full character table.
| maximal subgroup of | #13104331, #13104333, #13105824, … |
| maximal normal subgroup of | #13104331, #13104333, #13105824, … |
| maximal quotient of | #13105117, #13105119, #13105129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1133);