Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#499867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C25 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
17017 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | #72266741, #72266750, #72267282, … |
| maximal normal subgroup of | #72266741, #72266750, #72267282, … |
| maximal quotient of | #72266741, #72266750, C2×#72210362 |
Past the SmallGroups library coverage — no SmallGroups id.