#72321149

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#151129
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72321047
prime radical	C₂×#2262	C₂×C₄
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

468 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)(3,19)(4,26)(5,31)(6,41)(7,44)(8,18)(9,17)(10,13)(11,20)(12,54)(14,36)(15,43)(16,34)(21,79)(22,63)(23,89)(24,57)(25,75)(27,68)(28,59)(29,78)(30,100)(32,105)(33,70)(35,90)(37,104)(38,116)(39,74)(40,101)(42,55)(45,77)(46,117)(47,120)(48,119)(49,76)(50,84)(51,110)(52,98)(53,85)(56,122)(58,107)(60,80)(61,94)(62,121)(64,96)(65,81)(66,126)(67,115)(69,82)(71,112)(72,111)(73,83)(86,124)(87,127)(88,93)(91,114)(92,97)(95,102)(99,123)(103,108)(106,128)(109,118)(113,125);(1,3)(2,11,13,50)(4,27)(5,32)(6,21)(7,45)(8,23)(9,29)(10,25)(12,55,56,15)(14,62,60,77)(16,69,61,105)(17,52,18,73)(19,31,75,101)(20,80)(22,34)(24,43)(26,90,79,116)(28,95)(30,47)(33,108)(35,106)(36,84)(37,107)(38,97)(39,109)(40,82)(41,53,68,59)(42,49)(44,51,119,70)(46,86)(48,121)(54,96)(57,103,76,91)(58,72,118,71)(63,102,93,125)(64,123,66,127)(65,124,67,100)(74,89,104,78)(81,83)(85,113)(87,111)(88,94)(92,117,128,120)(98,115)(99,112)(110,114)(122,126);(1,4,6,29,8,24,42,3,10,21,27,23,9,49,43,25)(2,12,15,52,17,59,68,50,13,56,55,73,18,53,41,11)(5,33,35,82,37,97,108,109,40,110,38,32,39,106,114,107)(7,22,34,86,46,87,111,121,48,88,94,30,47,99,112,45)(14,63,64,62,65,123,102,124,60,93,66,77,67,127,125,100)(16,70,44,118,71,128,120,69,61,51,119,58,72,92,117,105)(19,74,76,116,78,101,79,103,75,104,57,90,89,31,26,91)(20,81,28,96,83,36,54,113,84,115,85,126,98,80,122,95)

Polycyclic presentation:

2^11|b^2=d,c^2=f,e^2=gi,f^2=h,g^2=ij,h^2=k,i^2=jk,j^2=k,b^a=be,c^a=ceh,c^b=cdi,d^a=dk,e^a=eg,e^b=egk,e^c=egjk,f^a=fj,f^e=fk,g^a=gi,g^b=gi,g^c=gik,h^a=hk,i^a=ij,i^b=ij,i^c=ij,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321290, #72323074, #72323087, …
maximal normal subgroup of	#72321290, #72323074, #72323087, …

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