#72226893

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	#72225567	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

1539 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)(3,15)(4,21)(5,8)(6,12)(7,14)(9,19)(10,26)(11,22)(13,24)(16,28)(17,43)(18,44)(20,32)(23,55)(25,40)(27,31)(29,59)(30,57)(33,46)(34,58)(35,51)(36,49)(37,53)(38,48)(39,60)(41,56)(42,52)(45,50)(47,61)(54,62)(63,64);(3,16)(4,22)(5,24)(6,26)(8,13)(9,32)(10,12)(11,21)(15,28)(17,46)(18,47)(19,20)(23,56)(25,60)(27,59)(29,31)(30,34)(33,43)(35,63)(36,45)(37,62)(38,42)(39,40)(41,55)(44,61)(48,52)(49,50)(51,64)(53,54)(57,58);(2,8,10,29,14,31,12,13)(3,17,18,50)(4,23)(9,33,35,28,37,36,38,61)(11,40,39,21,41,30,34,55)(15,42,43,62,44,63,45,32)(16,46,47,49)(19,48,53,51)(20,52,54,64)(22,56)(25,57)(58,60);(1,3,4,19,24,52,58,49,6,17,25,51,27,54,56,47,7,18,23,53,59,64,60,46,26,50,57,48,5,20,22,16)(2,9,34,43,31,61,55,63,10,38,11,15,13,36,40,62,14,37,39,45,8,28,21,42,12,35,41,44,29,33,30,32)

Polycyclic presentation:

2^11|c^2=e,d^2=fg,e^2=i,f^2=hj,g^2=i,h^2=jk,j^2=k,c^a=ch,c^b=cg,d^a=df,d^b=dfghi,d^c=dj,e^a=ejk,e^b=ei,e^d=ek,f^a=fh,f^b=fh,f^c=fk,g^a=gj,g^b=gi,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, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
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, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
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);