Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#72225063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#442 |
| commutator | C22×C8 | C24×C4 |
| Frattini, cosocle | C2×C4×C8 | C25 |
| prime radical | C22×#2184 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ C22×#1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
12714 conjugacy classes of subgroups.
| transitive degree | μt(G) = 512 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72225133, #72322912 |
| maximal normal subgroup of | C2×#72225133, #72322912 |
| maximal quotient of | C2×#72225103, #72322912, C2×#72225085 |
Past the SmallGroups library coverage — no SmallGroups id.