Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| 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 | |
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:
(1,2,3,4)(5,6,9,7)(8,11,12,10)(15,19)(17,20);(6,7)(10,11)(13,14,16,18)(15,17,19,20);(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,19)(18,20);(5,8)(6,10)(7,11)(9,12)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #8737, #8755, #8758, … |
| maximal normal subgroup of | #8737, #8755, #8758, … |
| maximal quotient of | #13776, #13778, #13787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1045);