Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C22×#511771 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,15)(14,16);(1,3,5,4)(2,6,7,8)(11,16)(13,14);(1,4,5,3)(2,6,7,8)(10,13)(11,16)(14,15)Polycyclic presentation:
2^7|a^2=d,b^2=d,c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7443, #12234, #12236, … |
| maximal normal subgroup of | #7443, #12234, #12236, … |
| maximal quotient of | #10567, #10570, #10572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 792);