Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#512663 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2127 |
| commutator | C42 | C24×C4 |
| Frattini, cosocle | #127 | C25 |
| prime radical | C2×#60806 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×C4 ◁ C22×#535 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
6139 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
94 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213428, #72250935, C2×#72216601, … |
| maximal normal subgroup of | C2×#72213428, #72250935, C2×#72216601, … |
| maximal quotient of | #72285423 |
Past the SmallGroups library coverage — no SmallGroups id.