Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#41×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C27 |
| socle | C23 | C24×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
79533 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
220 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×#525, C2×AGL(1,ℤ/8)2, C2×#72242442 |
| maximal normal subgroup of | AGL(1,ℤ/8)×#525, C2×AGL(1,ℤ/8)2, C2×#72242442 |
| maximal quotient of | C2×#72249682, AGL(1,ℤ/8)×#525, #72305487, … |
Past the SmallGroups library coverage — no SmallGroups id.