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:
(1,2)(3,7)(4,5)(6,8)(10,14)(11,17)(12,20)(22,24);(2,5)(6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3,4,8)(2,6,5,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^7|b^2=g,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 | #8503, #18708, #18709, … |
| maximal normal subgroup of | #8503, #18708, #18709, … |
| maximal quotient of | #34373, #34383, #34393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2286);