Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(14,17)(15,19);(6,10)(7,11)(13,14,16,17)(15,18,19,20);(1,2)(3,4)(5,7,9,11)(6,8,10,12);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^7|a^2=f,b^2=eg,c^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9408, #9412, #9943, … |
| maximal normal subgroup of | #9408, #9412, #9943, … |
| maximal quotient of | #13923, #13965, #14043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1315);