#72227095

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227094
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#513051	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

819 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dhj,d^2=g,e^2=hk,f^2=ij,g^2=k,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cf,c^b=cefj,d^b=dj,d^c=di,e^a=eh,e^b=eh,e^c=ehj,e^d=ek,f^a=fj,f^b=fj,f^c=fij,g^b=gk,g^c=gk,h^a=hj,h^b=hj,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	#72227105, #72227138, #72227179
maximal normal subgroup of	#72227105, #72227138, #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.4*F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11);;
gap> SetPower(coll, 6, F.9*F.10);;
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);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
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);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
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);;
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);