Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C22×#33559 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C26 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
38421 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#7622880, C2×#72236949, #72337481, … |
| maximal normal subgroup of | C22×#7622880, C2×#72236949, #72337481, … |
| maximal quotient of | #72256208, #72313119, #72313133, … |
Past the SmallGroups library coverage — no SmallGroups id.