#72227109

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	#72227108
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

710 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dhj,c^2=dfi,d^2=g,e^2=hk,f^2=i,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cf,c^b=cefhj,d^b=dj,d^c=di,e^a=eh,e^b=eh,e^c=eh,e^d=ek,f^a=fk,f^b=fk,f^c=fi,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227121, #72227138, #72227179
maximal normal subgroup of	#72227121, #72227138, #72227179

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.8*F.10);;
gap> SetPower(coll, 3, F.4*F.6*F.9);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11);;
gap> SetPower(coll, 6, F.9);;
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);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
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);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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);