Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C217 |
| direct product | C2×#6370735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×#120 |
| commutator | C23 | C25×C4 |
| Frattini, cosocle | C24 | C26 |
| prime radical | C22×H(7,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
232221 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
232 conjugacy classes — view the full character table.
| maximal subgroup of | #72251050, #72256634, #72256649, … |
| maximal normal subgroup of | #72251050, #72256634, #72256649, … |
| maximal quotient of | #72251251, #72266870 |
Past the SmallGroups library coverage — no SmallGroups id.