Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215×C4 |
| direct product | C22×#25771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
69317 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
172 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D4×B(4,𝔽2), #72282628, #72282848 |
| maximal normal subgroup of | C22×D4×B(4,𝔽2), #72282628, #72282848 |
| maximal quotient of | #72303857, #72312976, #72313026, … |
Past the SmallGroups library coverage — no SmallGroups id.