#72323156

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#93757
commutator	C₃₂	C₄×C₁₆
Frattini, cosocle	C₂×#7330	C₂²
socle	C₂	#72286530
prime radical	C₂³	#7389
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=degij,c^2=f,d^2=gk,e^2=gk,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bd,c^b=cij,d^a=dg,d^b=dgijk,d^c=dj,e^a=egk,e^b=egijk,e^c=ej,f^b=fjk,f^d=fk,f^e=fk,g^a=gi,g^b=gijk,g^c=gk,h^b=hk,i^a=ij,i^b=ijk,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323158, #72323166, #72323173, …
maximal normal subgroup of	#72323158, #72323166, #72323173, …

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