Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210×C4 |
| direct product | C2×#494717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#99 |
| commutator | C22×C4 | C24×C4 |
| Frattini, cosocle | C22×D4 | C25 |
| prime radical | C2×#35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×D4 ◁ C23×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
19849 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72233119, #72256203, #72256415 |
| maximal normal subgroup of | C2×#72233119, #72256203, #72256415 |
| maximal quotient of | #72237490 |
Past the SmallGroups library coverage — no SmallGroups id.