#72227084

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, socle	C₂	#72227021
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#490484	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227091, #72227166, #72227179
maximal normal subgroup of	#72227091, #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, 3, F.4*F.6*F.9*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
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.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*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.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);;
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);