Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#7622880 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #386 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7482, #7484, #7486, … |
| maximal normal subgroup of | #7482, #7484, #7486, … |
| maximal quotient of | #8957, #8958, #8959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 616);