Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = #72236027 (order 2048) · Inn(G) = #121 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #121 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2,6,8,4,7,3,5);(2,7)(5,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)Polycyclic presentation:
2^7|a^2=g,b^2=df,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12390, #12391, #12414, … |
| maximal normal subgroup of | #12390, #12391, #12414, … |
| maximal quotient of | #8653, #9226, #9574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 426);