#72225320

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#11599
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225319
prime radical	C₂×#508	C₄²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,11)(3,16,21,51)(4,15,8,13)(5,25,29,39)(7,32,31,40)(9,34,35,19)(10,28,14,24)(12,42,41,30)(17,43,22,58)(18,54,20,50)(23,48,27,60)(26,61,62,36)(33,64,53,49)(37,55,47,56)(38,52,46,57)(44,59,45,63);(1,3,4,18,6,21,8,20)(2,9,11,35)(5,26,27,52,29,62,23,57)(7,33,24,22,31,53,28,17)(10,38,14,46)(12,43)(13,37,15,47)(16,48)(19,42,56,59,34,30,55,63)(25,50)(32,61,40,36)(39,54)(41,58)(44,49)(45,64)(51,60);(2,10,13,32,11,14,15,40)(3,17,20,53,21,22,18,33)(12,39,44,48,41,25,45,60)(19,52,56,62,34,57,55,26)

Polycyclic presentation:

2^11|a^2=k,b^2=d,c^2=f,d^2=g,e^2=hj,f^2=i,h^2=jk,j^2=k,b^a=be,c^a=chjk,c^b=chij,d^a=djk,e^a=eh,e^b=ehjk,e^d=ek,f^a=fj,f^b=fj,g^a=gk,h^a=hj,h^b=hjk,i^a=ik,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227037, #72227066, …
maximal normal subgroup of	#72225336, #72227037, #72227066, …

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