#72227101

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227100
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#513051	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1299 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=eij,d^2=gjk,e^2=h,f^2=ik,i^2=j,j^2=k,b^a=bf,c^a=cg,c^b=cfgj,d^b=dk,d^c=dgj,e^b=ej,e^c=ek,f^a=fi,f^b=fi,f^c=fij,f^e=fk,g^a=gk,g^b=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ijk,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227105, #72227158, #72227175
maximal normal subgroup of	#72227105, #72227158, #72227175

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