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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(19,20)(22,23);(2,5)(3,8)(9,10,12,14)(11,13,16,15)(17,18,21,24)(19,22,20,23);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^7|a^2=eg,b^2=ef,c^2=fg,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 | #7743, #7760, #7771, … |
| maximal normal subgroup of | #7743, #7760, #7771, … |
| maximal quotient of | #14446, #14464, #14467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1541);