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 | |
466 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,12)(15,16)(18,22)(19,25)(20,28)(30,32);(2,7)(4,8)(10,13)(12,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^7|b^2=g,c^2=fg,d^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 | #8529, #19059, #19060, … |
| maximal normal subgroup of | #8529, #19059, #19060, … |
| maximal quotient of | #34372, #34390, #34417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2285);