Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C2×#188216 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22×D4 |
| commutator | C22×C4 | C24×C4 |
| Frattini, cosocle | C2×C42 | C25 |
| socle | C24 | C22×C22⋊C4 |
| prime radical | C22×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
34569 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
208 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72225125, #72321798, #72322140, … |
| maximal normal subgroup of | C2×#72225125, #72321798, #72322140, … |
| maximal quotient of | C2×#72225044, #72321613, C2×#72319973, … |
Past the SmallGroups library coverage — no SmallGroups id.