#72323002

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, socle	C₂²	C₂×#12067
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂×#2262	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122214 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

834 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:

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

Polycyclic presentation:

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

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323032, #72323120, #72323147
maximal normal subgroup of	#72323032, #72323120, #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);;
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.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);