#72323142

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	#13661	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

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 512

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

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

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^b=dik,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	#72323144
maximal normal subgroup of	#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, 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, 2, F.4*F.9*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);