Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,7)(8,11,12,10)(14,15)(18,19);(2,4)(6,7)(8,12)(13,14,17,15)(16,19,20,18);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^7|a^2=ef,b^2=f,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #7742, #7758, #7781, … |
| maximal normal subgroup of | #7742, #7758, #7781, … |
| maximal quotient of | #14399, #14445, #14467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1465);