Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C4×#511790 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #32898 |
| commutator | C22×D4 | C24×C4 |
| Frattini, cosocle | C23×D4 | C25 |
| socle | C22 | C2×#32898 |
| prime radical | C2×#511790 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
At least 10979 conjugacy classes of subgroups; only the 1744 normal subgroups are shown.
212 conjugacy classes — view the full character table.
| maximal subgroup of | C4×AGL(2,ℤ/4), AGL(1,𝔽5)×#511790, C4×#72209772, … |
| maximal normal subgroup of | C4×B(5,𝔽2) |
| maximal quotient of | AGL(1,𝔽5)×#511790 |
| automorphism group of | C5×C4⋊Q8 |
Past the SmallGroups library coverage — no SmallGroups id.