Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,5,6)(3,7,8,4)(10,13)(14,16);(2,6)(4,7);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16);(1,4)(2,3)(5,7)(6,8);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|a^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8418, #17548, #17598, … |
| maximal normal subgroup of | #8418, #17548, #17598, … |
| maximal quotient of | #34194, #34233, #34235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2252);