#72227257

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#72225323
commutator	C₄×C₃₂	C₈²
Frattini, cosocle	#72216231	C₂²
socle	C₂	#72227255
prime radical	C₂³	#72214361
LCS	G ▷ C₄×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₃₂ ◁ C₄×M₆₄ ◁ C₈×#1398 ◁ #72216231 ◁ G
LECS	G ▷ #72216231 ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^13|a^2=c,b^2=dehjlm,c^2=f,d^2=gj,e^2=h,f^2=lm,g^2=im,h^2=j,i^2=k,j^2=l,k^2=m,l^2=m,b^a=bd,c^b=cijklm,d^a=dg,d^c=dkm,e^a=ejlm,e^c=el,f^b=fkl,f^d=fm,f^e=fm,g^a=gi,g^c=gm,h^a=hlm,h^c=hm,i^a=ik,j^a=jm,k^a=km

Conjugacy classes & character table

368 conjugacy classes — view the full character table.

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