#72227110

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225151
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#135	#106
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,12,6,14,26,46,9,16,28,15,8,38,7,11)(3,17,19,53,22,54,25,61,24,56,37,49,21,55,23,40)(5,29,32,50,33,45,34,51,36,48,27,62,35,64,31,13)(10,39,41,20,42,60,52,18,44,30,63,57,43,58,47,59);(1,3,4,25,6,21,26,19,9,24,28,23,8,22,7,37)(2,10,11,41,38,42,15,52,16,44,46,63,14,43,12,47)(5,30,32,59,33,60,34,57,36,39,27,18,35,58,31,20)(13,49,64,55,62,40,48,17,51,53,45,54,50,61,29,56);(2,11,16,46)(3,18,22,20,24,59,21,57)(4,26)(5,31,36,34)(6,8)(7,28)(10,40,42,49,44,61,43,53)(12,14,15,38)(17,41,55,52,56,63,54,47)(19,60,25,39,37,58,23,30)(27,35,32,33)(29,45)(48,64)(50,62)

Polycyclic presentation:

2^11|a^2=d,b^2=dfij,c^2=dfhij,d^2=g,e^2=h,f^2=ij,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cfhj,c^b=cehijk,d^b=djk,d^c=di,e^a=eh,e^b=eh,e^c=ehk,e^d=ek,f^a=fjk,f^b=fjk,f^c=fij,g^b=gk,g^c=gk,h^a=hj,h^b=hj,h^c=hj,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227121, #72227166, #72227171
maximal normal subgroup of	#72227121, #72227166, #72227171

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.9*F.10);;
gap> SetPower(coll, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);