#72322882

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

798 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:

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

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^a=ci,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322903, #72323133, #72323164
maximal normal subgroup of	#72322903, #72323133, #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);;
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, 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);