#72322779

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322776
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#367
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(1,2,6,14,44,69,105,29,52,73,5,13,37,48,100,99,49,72,45,108,36,53,109,111,28,31,38,106,8,15,46,70,9,18,47,35,64,32,39,66,10,16,40,43,110,33,101,103,7,17,41,67,12,61,63,104,50,71,42,68,51,4,27,30)(3,19,22,74,89,82,86,79,96,59,34,56,92,83,125,123,93,60,87,119,55,113,114,128,94,120,88,122,24,20,77,80,25,11,54,57,112,117,107,102,26,75,21,65,84,91,124,126,23,76,90,115,62,97,121,127,78,81,85,116,95,58,98,118);(1,3)(2,11)(4,20)(5,34)(6,22)(7,23)(8,24)(9,25)(10,26)(12,62)(13,65)(14,57)(15,58)(16,59)(17,60)(18,19)(21,40)(27,98)(28,94)(29,102)(30,80)(31,81)(32,82)(33,83)(35,74)(36,55)(37,92)(38,88)(39,107)(41,90)(42,85)(43,56)(44,89)(45,87)(46,77)(47,54)(48,91)(49,93)(50,78)(51,95)(52,96)(53,97)(61,113)(63,121)(64,112)(66,79)(67,119)(68,122)(69,117)(70,118)(71,120)(72,76)(73,75)(84,110)(86,105)(99,126)(100,125)(101,124)(103,123)(104,128)(106,116)(108,115)(109,114)(111,127);(1,4,28,32,7,48,52,61,9,15,50,69,49,33,10,53)(2,12,16,37,72,64,71,51,18,36,73,110,17,44,31,8)(3,20,78,117,23,91,26,97,25,58,94,82,93,83,96,113)(5,35,6,43,42,67,41,68,40,14,47,13,38,108,45,106)(11,55,75,92,76,89,81,95,19,62,59,84,60,112,120,24)(21,57,22,56,88,115,90,122,34,74,54,65,85,119,87,116)(27,99,63,111,101,70,39,66,46,103,109,104,100,30,105,29)(77,123,121,127,125,80,107,79,98,126,114,128,124,118,86,102)

Polycyclic presentation:

2^11|a^2=e,c^2=defg,d^2=gijk,e^2=gi,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bk,c^a=cd,c^b=ch,d^b=dk,d^c=dg,e^c=egjk,f^a=fj,f^c=fjk,f^d=fk,f^e=fk,g^c=gi,h^a=hk,h^c=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322837, #72323055, #72323105
maximal normal subgroup of	#72322837, #72323055, #72323105

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.4*F.5*F.6*F.7);;
gap> SetPower(coll, 4, F.7*F.9*F.10*F.11);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);