Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
149 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)(6,7)(9,10,13,11)(12,14,16,15);(5,6)(7,8)(9,11,13,10)(12,15,16,14);(2,4)(5,7,8,6)(10,14)(11,15)(12,16)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,f^b=fg,e^c=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7762, #7763, #7764, … |
| maximal normal subgroup of | #7762, #7763, #7764, … |
| maximal quotient of | #10406, #10407, #10414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 775);