Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23);(2,5)(6,8)(10,14)(11,18)(16,20)(19,24);(1,2,4,5)(3,6,7,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)Polycyclic presentation:
2^7|a^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8502, #18706, #18707, … |
| maximal normal subgroup of | #8502, #18706, #18707, … |
| maximal quotient of | #34367, #34383, #34394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2279);