Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#12711 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #1781 |
| commutator | C22×C4 | C22×C42 |
| Frattini, cosocle | C22×#41 | C24 |
| socle | C22 | C2×#1781 |
| prime radical | C2×#3484 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
8087 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#511819, #72283126, #72312460, … |
| maximal normal subgroup of | C4×#511819, #72283126, #72312460, … |
| maximal quotient of | #72283079 |
Past the SmallGroups library coverage — no SmallGroups id.