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 | |
256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,7)(8,12)(13,14,16,18)(15,17,20,19);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,18)(17,20);(5,8)(6,10)(7,11)(9,12)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9478, #9482, #9514, … |
| maximal normal subgroup of | #9478, #9482, #9514, … |
| maximal quotient of | #13870, #13958, #13992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1216);