#72227083

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225229
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#7623661	C₄
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

2066 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:

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

Polycyclic presentation:

2^11|a^2=d,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=cehj,d^b=dj,d^c=di,e^a=eh,e^b=ehj,e^c=eh,e^d=ek,f^a=fk,f^c=fi,g^b=gk,g^c=gk,h^a=hj,h^b=hjk,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	#72227091, #72227138, #72227171
maximal normal subgroup of	#72227091, #72227138, #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, 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.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*F.10);;
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, 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*F.11);;
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);