#72321979

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	#72225424	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

2^11|b^2=gi,d^2=f,e^2=gi,f^2=h,g^2=ij,h^2=jk,i^2=jk,j^2=k,b^a=bghijk,c^a=ce,c^b=cgik,d^a=dgj,d^b=dj,d^c=dgij,e^a=eg,e^b=ek,e^c=eg,e^d=ejk,f^a=fij,f^b=fk,f^c=fi,f^e=fk,g^a=gi,g^c=gi,g^d=gk,h^a=hj,h^c=hjk,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, 2, F.7*F.9);;
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, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*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.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*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);