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 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,15);(2,3)(6,7)(10,13)(14,16);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16);(1,4)(2,6)(3,7)(5,8)Polycyclic presentation:
2^7|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 | #8483, #8510, #18240, … |
| maximal normal subgroup of | #8483, #8510, #18240, … |
| maximal quotient of | #34377, #34382, #34398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2290);