#72323145

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	#513080	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122219 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

737 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:

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

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=di,d^c=di,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

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323147, #72323169
maximal normal subgroup of	#72323147, #72323169

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);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
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);