Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#149742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #11499 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | #379 | C24 |
| prime radical | C23×H(5,𝔽2) | C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×C4 ◁ C2×#2309 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
9713 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
82 conjugacy classes — view the full character table.
| maximal subgroup of | #72340267, C2×#72236934, C2×#72213786, … |
| maximal normal subgroup of | #72340267, C2×#72236934, C2×#72213786, … |
Past the SmallGroups library coverage — no SmallGroups id.