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 |
| prime radical | #121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,4,11,7)(5,13,14,8)(10,15,16,12)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^7|a^2=f,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 | #8501, #18710, #18711, … |
| maximal normal subgroup of | #8501, #18710, #18711, … |
| maximal quotient of | #34373, #34383, #34394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2287);