Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×#494583 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C24×C4 |
| Frattini, cosocle | C22×D4 | C25 |
| prime radical | C22×#3027 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
17153 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72233302, C2×C4×#33559, C2×#72260921, … |
| maximal normal subgroup of | C2×#72233302, C2×C4×#33559, C2×#72260921, … |
| maximal quotient of | #72316904, #72317002 |
Past the SmallGroups library coverage — no SmallGroups id.