#72322865

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	C₂×#2263	C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122214 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|a^2=e,c^2=egi,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chijk,d^c=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

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322903, #72323144, #72323147
maximal normal subgroup of	#72322903, #72323144, #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, 3, F.5*F.7*F.9);;
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, 2, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, 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);