#72225331

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#128688
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72216549
prime radical	C₂×#508	C₄²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#1313 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

859 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=f,c^2=d,d^2=g,e^2=hj,f^2=i,h^2=jk,i^2=k,j^2=k,b^a=be,c^a=cj,c^b=chi,d^a=dk,d^b=djk,e^a=eh,e^b=eh,e^c=ej,e^d=ek,f^c=fk,g^b=gk,h^a=hj,h^b=hj,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227167, #72227168
maximal normal subgroup of	#72225336, #72227167, #72227168

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