Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#72216330 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #12067 |
| commutator | C16 | C22×C4×C8 |
| Frattini, cosocle | C2×#346 | C24 |
| prime radical | C23×D16 | C2×C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ C22×#1313 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2604 conjugacy classes of subgroups.
188 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×AGL(1,ℤ/32), C2×#72225052, #72287081, … |
| maximal normal subgroup of | C2×C4×AGL(1,ℤ/32), C2×#72225052, #72287081, … |
| maximal quotient of | C2×#72223654, #72287081 |
Past the SmallGroups library coverage — no SmallGroups id.