#72227069

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227054
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#2246	C₂×C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

585 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=f,d^2=g,e^2=h,f^2=ik,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cfhj,c^b=cefhij,d^b=djk,d^c=di,e^a=eh,e^b=ehj,e^c=ehjk,e^d=ek,f^a=fk,f^c=fi,g^b=gk,g^c=gk,h^a=hj,h^b=hjk,h^c=hjk,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227077, #72227166, #72227179
maximal normal subgroup of	#72227077, #72227166, #72227179

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