#72321268

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	#151055
commutator	C₂²×C₃₂	C₂²×C₄
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72321083
prime radical	C₂⁴	#1426
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ C₂²×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=f,e^2=gijk,f^2=h,g^2=ijk,h^2=k,i^2=jk,j^2=k,b^a=bd,c^a=cei,c^b=cdhjk,d^c=dk,e^c=eg,f^a=fj,f^e=fk,g^c=gi,h^a=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321290, #72323081, #72323105
maximal normal subgroup of	#72321290, #72323081, #72323105

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