#72323184

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1333
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂³
prime radical	C₂⁴	#1333
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1052 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=efhjk,d^2=g,e^2=h,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,c^a=ce,c^b=cijk,d^c=dh,e^a=ejk,e^c=ehjk,e^d=ek,f^a=fjk,f^c=fhjk,f^d=fk,g^c=gj,h^a=hk,h^c=hjk,i^c=ik,j^c=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323190
maximal normal subgroup of	#72323190
maximal quotient of	#72323185

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