#72322659

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	#72322657
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂²×D₃₂	C₂×C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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:

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

Polycyclic presentation:

2^11|b^2=f,c^2=d,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=chi,d^b=dk,e^a=eg,e^b=ejk,e^c=eg,f^a=fk,f^c=fj,f^e=fk,g^a=gi,g^b=gk,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322689, #72323081, #72323087
maximal normal subgroup of	#72322689, #72323081, #72323087

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