Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 73728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15);(1,2,5,6)(3,7,4,8)(9,11,14,10)(12,16,13,15);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16);(1,3,5,4)(2,8,6,7)(9,13,14,12)(10,15,11,16);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^7|b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^b=df50 conjugacy classes — view the full character table.
| Fitting subgroup of | #85604, #85654, #11711051, … |
| maximal subgroup of | #8441, #12773, #17520, … |
| maximal normal subgroup of | #8441, #12773, #17520, … |
| maximal quotient of | #34129, #34155, C4×#550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2215);