Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C2×#72225022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#1333 |
| commutator | C16 | C24×C8 |
| Frattini, cosocle | #346 | C25 |
| prime radical | C2×#62635 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C2×C4×#131 ◁ C2×#33326 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
9790 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
260 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#72225087, C2×#72225111, C2×#72225031 |
| maximal normal subgroup of | C2×#72225087, C2×#72225111, C2×#72225031 |
| maximal quotient of | #72322541 |
Past the SmallGroups library coverage — no SmallGroups id.