Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| 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 | |
575 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)(3,7)(4,5)(6,8)(10,13)(11,15);(2,5)(6,7)(10,13)(14,16);(9,10,12,13)(11,14,15,16);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8500, #18704, #18705, … |
| maximal normal subgroup of | #8500, #18704, #18705, … |
| maximal quotient of | #34367, #34383, #34393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2278);