Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C22×#33392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#102 |
| commutator | C8 | C25×C4 |
| Frattini, cosocle | C2×C8 | C26 |
| prime radical | C23×#576 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ C23×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
39221 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
232 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#7622637, #72256967, #72257271, … |
| maximal normal subgroup of | C22×#7622637, #72256967, #72257271, … |
| maximal quotient of | C22×D4×#361, C2×#72265242, #72315974, … |
Past the SmallGroups library coverage — no SmallGroups id.