#72323159

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	#513082	C₄
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

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 96 points:

(1,2,5,10,7,11,17,13,6,9,20,25,21,26,31,29)(3,12,15,19,16,27,28,18,14,8,22,23,30,24,32,4)(33,34,37,45,39,44,51,74,41,48,40,66,36,49,57,47)(35,43,53,76,52,78,86,79,55,80,67,94,56,75,50,84)(38,60,63,83,64,42,68,71,65,70,59,46,58,72,61,81)(54,88,90,77,62,93,85,96,91,69,87,73,89,95,92,82);(1,3)(2,8)(4,13)(5,15)(6,14)(7,16)(9,12)(10,23)(11,24)(17,28)(18,29)(19,25)(20,22)(21,30)(26,27)(31,32)(34,42)(35,50)(36,39)(37,57)(38,61)(40,51)(43,73)(44,60)(45,46)(47,81)(48,72)(49,70)(52,67)(53,56)(54,89)(55,86)(58,63)(59,64)(62,91)(65,68)(66,83)(69,94)(71,74)(75,82)(76,88)(77,80)(78,96)(79,95)(84,93)(87,90);(1,4,17,27,21,23,5,12,6,18,31,24,7,19,20,8)(2,3,13,28,26,30,10,15,9,14,29,32,11,16,25,22)(33,35,51,86,36,56,37,53,41,55,57,50,39,52,40,67)(34,43,74,79,49,75,45,76,48,80,47,84,44,78,66,94)(38,62,68,87,58,54,63,85,65,89,61,90,64,91,59,92)(42,69,46,82,60,93,71,73,72,88,83,96,70,95,81,77);(2,9)(3,14)(10,25)(11,26)(13,29)(15,22)(16,30)(28,32)(33,36,41,39)(34,44,48,49)(35,52,55,56)(37,57,40,51)(38,58,65,64)(42,70,72,60)(43,75,80,78)(45,74,66,47)(46,81,83,71)(50,53,86,67)(54,62,91,89)(59,68,63,61)(69,93,88,95)(73,96,77,82)(76,84,94,79)(85,87,92,90)

Polycyclic presentation:

2^11|a^2=e,c^2=eg,d^2=k,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chjk,d^a=di,d^b=di,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	#72323164
maximal normal subgroup of	#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, 4, F.11);;
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, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
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);