#72321676

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286684
commutator	C₂×C₃₂	C₂²×C₈
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

512 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)(3,16)(4,23)(5,29)(6,26)(7,32)(9,39)(11,49)(12,24)(13,57)(14,61)(15,48)(17,72)(18,75)(19,76)(20,79)(21,80)(22,68)(25,66)(27,88)(28,42)(30,96)(31,87)(33,95)(34,93)(35,65)(36,92)(37,71)(38,83)(40,99)(41,105)(43,70)(44,50)(46,59)(47,56)(51,120)(52,119)(53,116)(54,64)(55,114)(58,63)(60,100)(62,106)(67,69)(73,117)(74,126)(77,102)(78,107)(82,91)(84,123)(85,124)(86,109)(89,118)(90,94)(97,113)(98,128)(101,121)(103,110)(104,108)(111,127)(112,115)(122,125);(1,2,6,10,8,15,26,48)(3,17)(4,24,23,81,9,12,39,45)(5,30,33,60,34,46,36,13)(7,40,38,109,42,14,62,47)(11,43)(16,64,68,123)(18,52)(19,63)(20,44)(21,54)(22,74)(25,53)(27,89,90,55,35,98,82,85)(28,56,83,61,32,86,106,99)(29,57,92,59,93,100,95,96)(31,101,103,127)(37,84)(41,111)(49,75,116,79)(50,70,119,66)(51,87)(58,102,117,104)(65,114,94,118,88,124,91,128)(67,121)(69,125,105,120)(71,72,80,126)(73,78)(76,112,107,97)(77,115)(108,113)(110,122);(1,3,9,43,26,21,23,18,8,22,4,25,6,37,39,20)(2,11,12,54,48,52,81,74,15,53,24,84,10,44,45,17)(5,31,27,76,33,105,90,104,34,103,35,107,36,69,82,102)(7,41,32,78,62,87,83,108,42,67,28,19,38,110,106,77)(13,58,85,127,30,97,89,125,60,117,55,101,46,112,98,120)(14,63,56,122,109,115,99,111,40,73,86,51,47,113,61,121)(16,65,75,93,71,94,70,95,68,88,79,29,80,91,66,92)(49,114,126,100,50,118,64,96,116,124,72,57,119,128,123,59)

Polycyclic presentation:

2^11|b^2=f,c^2=fhij,d^2=jk,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=cd,d^a=dj,d^b=dk,d^c=dj,e^a=eg,e^b=ejk,e^c=egj,f^a=fk,f^c=fjk,f^e=fk,g^a=gi,g^b=gk,g^c=gik,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	#72321722, #72323074, #72323081
maximal normal subgroup of	#72321722, #72323074, #72323081

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