#72286865

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#12067
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂×#12067
prime radical	C₂×#2263	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,10,14,12,7,11,16,4,3,8,13,15)(17,18,20,25,21,27,32,28,22,29,38,40,39,30,37,46)(19,24,33,26,34,23,31,41,36,42,35,45,48,43,47,44);(1,3,7,6)(2,8,11,10)(4,12,9,15)(5,13,16,14);(1,3,7,6)(2,8,11,10)(4,12,9,15)(5,13,16,14)(18,23)(19,31)(20,37)(21,39)(24,27)(25,45)(26,40)(28,41)(29,43)(30,42)(32,38)(33,34)(35,48)(36,47)(44,46);(1,4,14,2,3,12,5,8,7,9,13,11,6,15,16,10)(17,19,32,31,39,48,20,33,22,36,37,47,21,34,38,35)(18,24,28,41,30,43,25,26,29,42,46,44,27,23,40,45)

Polycyclic presentation:

2^11|a^2=e,b^2=ik,c^2=ik,d^2=eg,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=cf,d^c=dhjk,e^c=ejk,f^a=fh,f^c=fhjk,f^d=fj,f^e=fk,g^c=gk,h^a=hj,h^c=hjk,h^d=hk,j^a=jk,j^c=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287443, #72322953, #72323120, …
maximal normal subgroup of	#72287443, #72322953, #72323120, …
maximal quotient of	#72287331

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