Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,12)(8,14)(9,13)(10,15)(11,16)(17,20)(18,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,13)(14,20)(15,19);(1,3,4,2)(5,7)(6,12)(8,18)(9,16)(10,17)(11,13)(14,19)(15,20);(5,8,10,9)(6,13,15,14)(7,16,17,18)(11,12,19,20)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=g,e^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12945, #12946, #13022, … |
| maximal normal subgroup of | #12945, #12946, #13022, … |
| maximal quotient of | #17857, #17881, #17898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1683);