#72227147

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225051
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512851	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479908 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2031 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:

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

Polycyclic presentation:

2^11|a^2=g,d^2=e,e^2=h,f^2=ij,g^2=k,i^2=jk,j^2=k,b^a=bf,c^a=cgjk,c^b=cfijk,d^a=dk,d^b=dijk,e^b=ejk,f^a=fi,f^b=fi,f^c=fi,f^d=fj,f^e=fk,g^c=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ij,i^d=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227158, #72321568
maximal normal subgroup of	#72227158, #72321568

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