#72225361

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	C₄	#138740
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72216313
prime radical	#512928	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₄×#1313 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=eh,c^2=d,d^2=g,e^2=hj,f^2=i,h^2=jk,j^2=k,b^a=be,c^a=chj,c^b=cf,d^a=djk,d^b=di,e^a=eh,e^c=ej,e^d=ek,f^a=fjk,f^b=fij,g^a=gk,h^a=hj,h^c=hk,i^a=ik,i^b=ik,j^a=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225374, #72227138, #72227166
maximal normal subgroup of	#72225374, #72227138, #72227166

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