#72287393

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	#513003	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

799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|a^2=e,b^2=ik,c^2=i,d^2=ik,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=fh,f^e=fk,g^d=gk,h^a=hk,h^c=hj,h^d=hj,j^c=jk,j^d=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287443, #72323133
maximal normal subgroup of	#72287443, #72323133

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.9*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);;
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);;
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);