#72323102

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	C₂×C₄	#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂×#12067
prime radical	C₂×#2262	C₈
LCS	G ▷ C₂×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

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,5,9,7,8,14,4,6,10,16,13,3,11,12,15)(17,18)(19,29,32,57,31,58,44,60)(20,34,37,70)(21,39)(22,43)(23,25)(24,33,47,69,48,65,54,66)(26,38,49,75)(27,42)(28,45)(30,56,63,59,62,51,40,78)(35,53,72,46,68,80,64,76)(36,55,73,71)(41,77)(50,61)(52,67,79,74);(1,3,6,7)(2,8,10,11)(4,13,15,9)(5,12,16,14)(17,19)(18,24)(20,35)(21,40)(22,44)(23,31)(25,48)(26,51)(27,53)(28,54)(29,55)(30,61)(32,45)(33,67)(34,57)(36,72)(37,68)(38,69)(39,76)(41,63)(42,62)(43,47)(46,77)(49,56)(50,80)(52,78)(58,71)(59,79)(60,70)(64,73)(65,74)(66,75);(1,4,14,11,3,9,5,10,6,15,12,8,7,13,16,2)(19,30,44,40,31,62,32,63)(20,36,37,73)(21,41)(24,46,54,53,48,76,47,80)(26,52,49,79)(27,50)(29,56,60,78,58,51,57,59)(33,68,66,72,65,35,69,64)(34,55,70,71)(38,67,75,74)(39,77)(42,61);(18,25)(19,31)(26,49)(27,50)(28,43)(30,62)(32,44)(33,65)(34,70)(35,68)(39,77)(40,63)(52,79)(55,71)(64,72)(66,69)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323108, #72323133, #72323147, …
maximal normal subgroup of	#72323108, #72323133, #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);;
gap> SetPower(coll, 3, 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, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*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);