#72321695

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	#72286713
commutator	C₂×C₃₂	C₂²×C₈
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

503 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=hk,c^2=fik,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,c^a=ce,c^b=cd,d^b=dk,e^a=eg,e^c=eg,f^c=fjk,f^e=fk,g^a=gi,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321722, #72323105, #72323131
maximal normal subgroup of	#72321722, #72323105, #72323131

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