Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×#12707 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #1781 |
| commutator | C22×C4 | C22×C42 |
| Frattini, cosocle | C23×D4 | C24 |
| socle | C22 | C2×#1781 |
| prime radical | C2×#3484 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
8439 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#511790, AGL(1,𝔽5)×#12707, C4×#144378, … |
| maximal normal subgroup of | C4×#511790, C4×#144378, C4×#139743, … |
| maximal quotient of | AGL(1,𝔽5)×#12707 |
Past the SmallGroups library coverage — no SmallGroups id.