Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(9,10)(11,13)(12,14)(15,16);(5,6,7,8)(9,10,12,14)(11,13,16,15);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16);(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^7|a^2=f,b^2=g,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2657⋊C3, #11711254, #11711256, … |
| maximal subgroup of | #10088, #10096, #18426, … |
| maximal normal subgroup of | #10088, #10096, #18426, … |
| maximal quotient of | #14272, #14273, #14321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1389);