#72323181

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#7848
commutator	C₂×C₈	C₂×C₄×C₁₆
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72322703
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=f,d^2=g,e^2=j,f^2=h,g^2=i,h^2=j,i^2=j,j^2=k,b^a=bhij,c^a=ce,c^b=chij,d^b=dk,d^c=dj,e^a=ek,e^b=ek,f^a=fj,f^d=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323182
maximal normal subgroup of	#72323182

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