#72343256

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72343249
commutator	C₂²×C₁₆	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	C₂×#3415	C₂×C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #479960 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1627 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=fgijk,d^2=h,e^2=ik,f^2=i,g^2=i,i^2=j,j^2=k,b^a=bhik,c^a=ce,c^b=cf,d^b=dk,d^c=dj,e^a=ei,e^b=eijk,e^c=ehijk,e^d=ek,f^a=fik,f^b=fijk,f^c=fijk,f^d=fk,g^a=ghi,g^b=gijk,g^c=gijk,g^d=gk,g^e=gk,h^c=hk,i^a=ij,i^b=ijk,i^c=ijk,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343265, #72343385, #72343423, …
maximal normal subgroup of	#72343265, #72343385, #72343423, …

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