#72322773

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322759
commutator	C₂×C₃₂	C₂³×C₄
Frattini, cosocle	C₂²×C₃₂	C₂⁴
prime radical	#72232017	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=f,c^2=efh,d^2=g,e^2=hi,f^2=hi,h^2=ij,i^2=jk,j^2=k,b^a=bk,c^a=ce,c^b=cg,d^a=dj,d^c=djk,e^b=ek,e^c=eh,e^d=ek,f^c=fh,f^d=fk,g^a=gk,g^c=gk,h^c=hi,i^c=ij,j^c=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322837, #72322999, #72323123
maximal normal subgroup of	#72322837, #72322999, #72323123

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