#72287134

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#12067
commutator	C₂×C₁₆	C₂²×C₁₆
Frattini, cosocle	C₂×#1311	C₂³
prime radical	C₂²×D₃₂	C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#7053 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=i,c^2=ehk,d^2=g,e^2=hj,f^2=jk,g^2=i,h^2=jk,j^2=k,c^a=cf,c^b=ce,d^c=djk,e^a=ej,e^b=eh,e^d=ek,f^a=fk,f^b=fj,g^c=gk,h^a=hk,h^b=hj,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287331
maximal normal subgroup of	#72287331
maximal quotient of	#72287138

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