#72226961

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72214276
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
socle	C₂	#72225218
prime radical	#72225587	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|c^2=e,d^2=fh,f^2=hj,g^2=ik,h^2=j,i^2=kl,k^2=l,b^a=bg,c^a=ck,c^b=cjk,d^a=dg,d^b=df,d^c=dj,e^a=el,e^b=el,f^a=fi,f^b=fh,g^a=gi,g^b=gi,g^c=gl,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982
maximal normal subgroup of	#72226982

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
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.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
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.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);