Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 8192 |
| 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 | |
570 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)(3,4)(5,6)(7,8)(11,15)(13,16);(2,6)(3,7)(9,10,12,14)(11,13,15,16);(9,11,12,15)(10,13,14,16);(1,3,5,7)(2,4,6,8);(1,4)(2,7)(3,6)(5,8)Polycyclic presentation:
2^7|b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8358, #8430, #8434, … |
| maximal normal subgroup of | #8358, #8430, #8434, … |
| maximal quotient of | #34186, #34206, #34213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2238);