#72225601

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225600
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂³	#12438
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=d,c^2=fhik,e^2=gik,f^2=hjk,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=befhk,c^a=cfik,c^b=cfijk,d^c=dk,e^b=eg,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226838, #72226936, #72226950
maximal normal subgroup of	#72226838, #72226936, #72226950

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