#72235265

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#119556
commutator	C₂²×#121	C₂²×C₄
Frattini, cosocle	C₂²×#560	C₂³
prime radical	#5025789	C₂²
LCS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#560 ◁ G
LECS	G ▷ C₂²×#560 ▷ C₂⁴ ▷ C₂² ▷ C₁

2682 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)(3,17,20,4)(5,30,25,46)(6,14,35,13)(7,19,37,60)(8,16)(9,15)(10,21,41,31)(11,43)(12,53,54,33)(18,40,36,58)(22,26,34,27)(23,28,24,29)(32,55)(38,39,47,61)(42,52)(44,48,45,49)(50,59)(51,57)(56,62,64,63);(1,3,18,11,32,20,36,50)(2,10,40,27,55,44,58,39)(4,13,57,54,17,60,42,30)(5,31,35,47,53,49,37,22)(6,34,25,48,7,38,33,21)(8,24,62,52,9,23,63,51)(12,28,14,59,46,29,19,43)(15,45,56,26,16,41,64,61);(1,4,25,45,9,29,5,10)(2,11,46,34,15,51,30,22)(3,19,48,64,23,13,31,58)(6,27,63,42,37,26,18,59)(7,39,62,57,35,61,36,43)(8,28,53,44,32,17,33,41)(12,38,16,52,54,47,55,50)(14,21,56,24,60,49,40,20)

Polycyclic presentation:

2^11|a^2=d,b^2=def,c^2=defgik,e^2=jk,g^2=j,b^a=be,c^a=cefh,c^b=cdfgj,d^b=dh,d^c=di,e^a=ehjk,e^b=eik,e^c=ehk,e^d=ej,f^a=fhik,f^b=fhik,f^c=fj,f^d=fk,f^e=fjk,g^a=ghijk,g^b=ghj,g^c=ghik,g^d=gk,g^f=gj,h^a=hj,h^b=hjk,h^c=hk,i^a=ijk,i^b=ik,i^c=ij

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235272
maximal normal subgroup of	#72235272

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