Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C2×#7622451 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×#99 |
| commutator | C23 | C25×C4 |
| Frattini, cosocle | C2×D4 | C26 |
| prime radical | C2×#62842 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ C23×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
41093 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
178 conjugacy classes — view the full character table.
| maximal subgroup of | #72256620, C2×#72251110, C2×#72234840, … |
| maximal normal subgroup of | #72256620, C2×#72251110, C2×#72234840, … |
| maximal quotient of | #72263641, #72312192, #72312966, … |
Past the SmallGroups library coverage — no SmallGroups id.