#72323157

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	#13697	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

729 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)(33,34,37,46,39,48,56,79,41,45,61,49,42,71,40,70)(35,50,36,44,53,80,54,81,51,91,57,82,55,90,59,86)(38,62,65,87,66,96,67,83,68,85,60,78,69,73,95,47)(43,72,74,52,75,63,89,93,77,64,76,92,88,58,84,94);(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)(33,35)(34,43)(36,56)(37,54)(38,63)(39,55)(40,57)(41,51)(42,53)(44,78)(45,77)(46,84)(47,86)(48,88)(49,89)(50,73)(52,60)(58,68)(59,61)(62,90)(64,69)(65,92)(66,72)(67,93)(70,74)(71,75)(76,79)(80,85)(81,83)(82,87)(91,96)(94,95);(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)(33,36,56,51,42,59,37,53,41,57,40,35,39,54,61,55)(34,44,79,91,71,86,46,80,45,82,70,50,48,81,49,90)(38,64,67,94,69,63,65,92,68,72,95,93,66,58,60,52)(43,73,89,87,88,85,74,47,77,96,84,78,75,62,76,83);(2,9)(3,14)(10,25)(11,26)(13,29)(15,22)(16,30)(28,32)(34,45)(35,51)(36,57)(46,49)(47,83)(48,71)(52,92)(53,55)(54,59)(58,63)(62,85)(64,72)(70,79)(73,96)(78,87)(93,94)

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^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, 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);