Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = #72268601 (order 4096) · Inn(G) = C25 · Out(G) = C22×H(5,𝔽2) |
| 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 | |
456 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,23,26)(19,21,24,20)(22,30,31,25)(27,28,32,29);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,21,23,20)(18,19,26,24)(22,29,31,28)(25,27,30,32);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=f,d^2=g,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 | #8485, #8512, #18246, … |
| maximal normal subgroup of | #8485, #8512, #18246, … |
| maximal quotient of | #34378, #34382, #34399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2295);