Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C5 |
| direct product | Q8×#1039 |
| Sylow | C52 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C5⋊D5 |
| commutator | C2×C52 | C22×C4 |
| Fitting | C2×C52×Q8 | C2 |
| socle, prime radical, cosocle | C22×C52 | C23 |
| hypercenter | C2×Q8 | C5⋊D5 |
| hypocenter | C52 | C4×Q8 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
280 conjugacy classes of subgroups.
| transitive degree | μt(G) = 800 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | Q8⋊#6033, Q8⋊#6034, #13101738, … |
| maximal normal subgroup of | Q8⋊#6033, Q8⋊#6034, #13101738, … |
| maximal quotient of | C4⋊C4×#1039, #13102838, #13102842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 924);