Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#72224984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#2177 |
| commutator | C2×C16 | C24×C4 |
| Frattini, cosocle | C22×C16 | C25 |
| prime radical | C2×#512827 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C23×C4×C8 ◁ C23×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
12294 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72225052, #72287029, #72287579, … |
| maximal normal subgroup of | C2×#72225052, #72287029, #72287579, … |
| maximal quotient of | #72287579, #72323148 |
Past the SmallGroups library coverage — no SmallGroups id.