#72323141

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	#513009	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

769 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:

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

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^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, 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);