#72225307

order	8192
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#94857
commutator	C₈²	C₄²×C₈
Frattini, cosocle	#72216182	C₂³
socle	C₂²	#72216186
prime radical	C₂³	#72214029
LCS	G ▷ C₈² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ #72216182 ◁ G
LECS	G ▷ #72216182 ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1719 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^13|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=j,h^2=j,i^2=k,j^2=l,k^2=m,b^a=bhjl,c^a=cikm,d^b=djl,d^c=dkm,e^a=ejl,e^d=el,f^a=fkm,f^d=fm,g^b=gl,g^c=gm,h^a=hl,i^a=im

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225329
maximal normal subgroup of	#72225329

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.11*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.13);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.13);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);