Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C216 |
| direct product | C22×#21005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#121 |
| commutator | C2×C4 | C25×C4 |
| Frattini, cosocle | C22×C4 | C26 |
| prime radical | C24×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
66117 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
232 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72252147, C2×#72252143, C2×#72252144, … |
| maximal normal subgroup of | C2×#72252147, C2×#72252143, C2×#72252144, … |
| maximal quotient of | C2×#72308356, C2×#72308938, C2×#72309036 |
Past the SmallGroups library coverage — no SmallGroups id.