#72323018

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

799 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=i,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=chjk,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

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323032, #72323133, #72323144
maximal normal subgroup of	#72323032, #72323133, #72323144

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*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.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);