Finite Group DatabasebetaFinite Group Databasebeta| order | 27648 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 36528696852480 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C214×C3×C4 |
| abelian | C24×C3×C4×C9×C16 |
| multiplicative | (ℤ/90440)ˣ, (ℤ/100776)ˣ, (ℤ/105672)ˣ, (ℤ/108528)ˣ, (ℤ/116280)ˣ, (ℤ/127680)ˣ, (ℤ/128520)ˣ, (ℤ/135660)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3×C8 | C26×C32 |
| socle, prime radical | C26×C32 | C2×C3×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
254930 conjugacy classes of subgroups.
27648 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.