Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C24×#2846 (order 2048) · Inn(G) = C23 · Out(G) = C25×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C22×C8 |
| socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
130 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)(9,10,13,11,12,14,16,15);(2,4)(9,11,13,14,12,15,16,10);(5,6,7,8)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=e,f^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| maximal subgroup of | #7118, #8843, #8844, … |
| maximal normal subgroup of | #7118, #8843, #8844, … |
| maximal quotient of | #9593, #9594, #9595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 671);