Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219×C4 |
| direct product | C23×#3151 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C27 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C24×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
170357 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
256 conjugacy classes — view the full character table.
| maximal subgroup of | C23×D4×AGL(1,ℤ/8) |
| maximal normal subgroup of | C23×D4×AGL(1,ℤ/8) |
| maximal quotient of | C22×#6467620 |
Past the SmallGroups library coverage — no SmallGroups id.