#72227096

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225244
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#513051	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

973 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dfijk,d^2=g,e^2=h,f^2=ijk,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cfk,c^b=cehjk,d^b=djk,d^c=di,e^a=eh,e^b=eh,e^c=ehjk,e^d=ek,f^a=fjk,f^b=fjk,f^c=fijk,g^b=gk,g^c=gk,h^a=hj,h^b=hj,h^c=hjk,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227105, #72227166, #72227171
maximal normal subgroup of	#72227105, #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*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10*F.11);;
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.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*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.10*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*F.11);;
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*F.11);;
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);