Finite Group DatabasebetaFinite Group Databasebeta| order | 27648 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 521838526464 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C27×C3×C43 |
| abelian | C22×C3×C42×C9×C16 |
| multiplicative | (ℤ/62985)ˣ, (ℤ/65416)ˣ, (ℤ/66045)ˣ, (ℤ/67184)ˣ, (ℤ/70448)ˣ, (ℤ/79040)ˣ, (ℤ/82880)ˣ, (ℤ/83980)ˣ, (ℤ/88060)ˣ, (ℤ/90576)ˣ, (ℤ/92352)ˣ, (ℤ/95472)ˣ, (ℤ/98124)ˣ, (ℤ/106560)ˣ, (ℤ/112320)ˣ, (ℤ/113220)ˣ, (ℤ/119340)ˣ, (ℤ/125970)ˣ, (ℤ/132090)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C3×C8 | C25×C32 |
| socle, prime radical | C25×C32 | C22×C3×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
91410 conjugacy classes of subgroups.
27648 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.