Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
274 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)(5,6,9,8)(7,11,12,10)(14,17)(18,19);(6,8)(10,11)(13,14,16,17)(15,19,20,18);(5,7)(6,10)(8,11)(9,12);(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^7|a^2=g,b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| maximal subgroup of | #9257, #9261, #22513, … |
| maximal normal subgroup of | #9257, #9261, #22513, … |
| maximal quotient of | C22⋊C4×C4⋊C4, #13802, #13812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1077);