#72227145

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	#72227144
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512849	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #128258 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1504 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=g,d^2=e,e^2=h,f^2=ij,g^2=k,i^2=jk,j^2=k,b^a=bf,c^b=cfgik,d^a=djk,d^b=dik,d^c=dgjk,e^a=ek,e^b=ej,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

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227158
maximal normal subgroup of	#72227158

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