#72287413

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#513082	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122219 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=ik,c^2=i,d^2=ehijk,e^2=g,f^2=hj,g^2=i,h^2=jk,j^2=k,b^a=bik,c^b=cik,d^a=dhik,d^c=df,e^d=ejk,f^a=fj,f^c=fh,f^d=fhjk,f^e=fk,g^d=gk,h^a=hk,h^c=hj,h^d=hjk,j^c=jk,j^d=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287443, #72323147
maximal normal subgroup of	#72287443, #72323147

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