#72340473

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#151342
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72285472	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁶ ◁ C₂×#1781 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=i,b^2=ghij,d^2=hj,e^2=hj,f^2=k,b^a=bf,c^a=ck,c^b=ck,d^a=de,d^b=dik,e^a=eh,e^b=eg,e^d=ehj,f^b=fik,f^d=fgik,f^e=fijk,g^a=gi,g^b=gjk,g^d=gjk,g^e=gj,h^a=hk,h^b=hj,h^f=hk,i^b=ik,i^d=ij,i^e=ik,j^a=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340478
maximal normal subgroup of	#72340478

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