#72340939

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237744
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72285341	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20585 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

2148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=gijk,b^2=ghj,c^2=h,f^2=gjk,b^a=be,c^a=cf,c^b=cghik,d^a=djk,d^c=dghj,e^a=ej,e^b=ei,e^c=eh,f^a=fg,f^b=fhj,f^c=fgij,f^d=fijk,f^e=fijk,g^a=gk,g^b=gjk,g^c=gk,g^d=gk,g^e=gk,h^a=hi,h^b=hjk,h^d=hk,h^f=hjk,i^b=ik,i^c=ij,i^f=ik,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341003
maximal normal subgroup of	#72341003

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