Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | #41×#361 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#102 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| socle | C22 | C23×#102 |
| prime radical | C2×D8×#41 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ C2×C8×#41 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
6143 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×#361, #41×AGL(1,ℤ/16), #72342664 |
| maximal normal subgroup of | AGL(1,ℤ/8)×#361, #41×AGL(1,ℤ/16), #72342664 |
Past the SmallGroups library coverage — no SmallGroups id.