#72226902

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226892
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	#72225571	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #128674 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1537 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:

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

Polycyclic presentation:

2^11|a^2=g,c^2=e,d^2=fgh,e^2=i,f^2=hj,g^2=ij,h^2=jk,j^2=k,b^a=bgh,c^a=ch,c^b=cgh,d^a=df,d^b=dfgijk,d^c=dj,e^a=ejk,e^b=ei,e^d=ek,f^a=fh,f^b=fh,f^c=fk,g^b=gijk,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226910
maximal normal subgroup of	#72226910

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