Finite Group DatabasebetaFinite Group Databasebeta| order | 7168 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 8052548283924480 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C220×C4 |
| abelian | C25×C4×C7×C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C4 | C27×C7 |
| socle, prime radical | C27×C7 | C2×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
408234 conjugacy classes of subgroups.
7168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C25×C4×C7×C16, C25×C7×C82, C24×C42×C7×C8, … |
| maximal normal subgroup of | C25×C4×C7×C16, C25×C7×C82, C24×C42×C7×C8, … |
| maximal quotient of | C25×C4×C7×C16, C25×C7×C82, C24×C42×C7×C8, … |
Past the SmallGroups library coverage — no SmallGroups id.