#72225510

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225509
commutator	C₄×C₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₆₄	C₂⁴
prime radical	#72225430	C₂
LCS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×#346 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₄×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1529 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:

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

Polycyclic presentation:

2^12|a^2=l,c^2=f,d^2=g,e^2=hj,g^2=i,h^2=jk,j^2=kl,k^2=l,b^a=be,c^a=ck,c^b=cg,d^a=dj,d^b=dgi,e^a=eh,e^b=eh,e^c=ekl,f^a=fl,f^b=fi,f^e=fl,g^a=gkl,g^b=gi,h^a=hj,h^b=hj,h^c=hl,i^a=il,j^a=jk,j^b=jk,k^a=kl,k^b=kl

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550
maximal normal subgroup of	#72225550

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, 1, F.12);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
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);