#72323007

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×#2262	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

776 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:

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

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^a=ci,c^b=chijk,d^b=dk,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323032, #72323152, #72323164
maximal normal subgroup of	#72323032, #72323152, #72323164

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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*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);