#72322678

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322657
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#2262	C₂×C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=fjk,c^2=d,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=chi,d^b=dk,e^a=eg,e^b=ejk,e^c=egjk,f^c=fjk,f^e=fk,g^a=gi,g^b=gk,g^c=gik,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322689, #72323128, #72323131
maximal normal subgroup of	#72322689, #72323128, #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.6*F.10*F.11);;
gap> SetPower(coll, 3, F.4);;
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, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10*F.11);;
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, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*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);