Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C5 |
| direct product | Q8×#1044 |
| Fitting | C52×Q8 ⋊ C4 |
| Sylow | C52 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#1044 |
| commutator, socle | C2×C52 | C22×C4 |
| Fitting | C52×Q8 | C4 |
| prime radical, cosocle | C2×C5⋊D5 | C23 |
| hypercenter | Q8 | #1044 |
| hypocenter | C52 | C4×Q8 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
178 conjugacy classes of subgroups.
| transitive degree | μt(G) = 80 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | #13104075, #13104076, #13104079, … |
| maximal normal subgroup of | #13104075, #13104076, #13104079, … |
| maximal quotient of | #13104844, #13104845, #13104846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 1114);