Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C22×#6342371 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator, socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(6,8)(9,10,14,11,13,15,12,16);(2,4)(9,11,14,15,13,16,12,10);(5,6,7,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=e,f^2=g,b^a=bd,c^a=ce,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7124, #8878, #8879, … |
| maximal normal subgroup of | #7124, #8878, #8879, … |
| maximal quotient of | #9597, #9598, #9599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 672);