Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C2×#495677 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | #120 |
| commutator | C23 | C24×C8 |
| Frattini, cosocle | C22×C8 | C25 |
| socle | C22 | C8×#120 |
| prime radical | C2×#3025 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
6581 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
256 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72319766, C2×#72224961 |
| maximal normal subgroup of | C2×#72224961 |
| automorphism group of | C2×M128 |
Past the SmallGroups library coverage — no SmallGroups id.