#72321964

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225501
commutator	C₂×C₃₂	C₂³×C₄
Frattini, cosocle	#1397	C₂⁴
prime radical	#153607	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ #149495 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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)(3,16)(4,23)(5,31)(6,36)(7,10)(8,12)(9,19)(11,28)(13,27)(14,34)(15,38)(17,59)(18,64)(20,60)(21,71)(22,58)(24,74)(25,54)(26,77)(29,94)(30,95)(32,49)(33,98)(35,102)(37,57)(39,47)(40,62)(41,65)(42,69)(43,85)(44,84)(45,92)(46,91)(48,80)(50,115)(51,87)(52,79)(53,101)(55,86)(56,68)(61,119)(63,113)(66,120)(67,105)(70,116)(72,111)(73,124)(75,89)(76,93)(78,117)(81,127)(82,118)(83,108)(88,126)(90,97)(96,106)(99,109)(100,114)(103,121)(104,125)(107,123)(110,122)(112,128);(3,17)(4,24)(9,40)(11,48)(18,61)(20,22)(21,63)(23,74)(26,30)(28,80)(29,81)(32,72)(37,66)(41,103)(42,56)(43,104)(44,105)(45,51)(50,112)(54,117)(70,73)(75,110)(76,126)(77,95)(87,92)(88,93)(89,122)(94,127)(99,100)(106,108)(107,118)(115,128);(1,3)(2,9)(4,25)(5,32)(6,37)(7,20)(8,17)(10,42)(11,49)(12,40)(13,54)(14,41)(15,56)(16,23)(18,34)(19,28)(21,27)(22,38)(24,78)(26,83)(29,82)(30,96)(31,43)(33,72)(35,99)(36,44)(39,66)(45,109)(46,106)(47,105)(48,111)(50,116)(51,114)(52,117)(53,103)(55,108)(57,94)(58,77)(59,74)(60,95)(61,101)(62,80)(63,79)(64,76)(65,115)(67,122)(68,87)(69,92)(70,91)(71,75)(73,86)(81,123)(84,89)(85,93)(88,125)(90,100)(97,118)(98,104)(102,107)(110,113)(112,124)(119,126)(120,127)(121,128);(2,10,12,15)(3,18,20,66,17,61,22,37)(4,26,24,30)(5,33)(9,41,42,105,40,103,56,44)(11,48)(13,55,52,46)(14,47,53,36)(16,57,58,119,59,120,60,64)(19,67,68,65,62,84,69,121)(21,72,70,100,63,32,73,99)(25,82,83,125,78,123,96,85)(27,79)(28,92,80,87)(29,88,81,93)(31,97,98,102)(35,90)(43,108,107,54,104,106,118,117)(45,51)(49,113,114,116,111,71,109,124)(50,112)(75,110)(86,91)(89,115,122,128)

Polycyclic presentation:

2^11|d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bh,c^a=ce,c^b=ck,d^a=dj,d^c=dij,e^a=eg,e^b=ek,e^c=eg,e^d=ejk,f^a=fk,f^c=fj,f^e=fk,g^a=gi,g^c=gi,g^d=gk,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322015
maximal normal subgroup of	#72322015

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