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 | C2 | D5×C22⋊D5 |
| commutator, cosocle | C4×C52 | C23 |
| Frattini | C4 | C2×D52 |
| Fitting | C52×Q8 | C22 |
| socle | C2×C52 | C2×D4 |
| hypercenter | Q8 | D52 |
| hypocenter | C52 | #135 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
144 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 40 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^2=e,b^a=be,c^a=ce,g^a=g^4,c^b=cd,d^b=de,f^b=f^4,d^c=de68 conjugacy classes — view the full character table.
| maximal subgroup of | #13103757, #13103758, #13103761, … |
| maximal normal subgroup of | #13103757, #13103758, #13103761, … |
| maximal quotient of | #13100600, #13100611, #13100622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 726);