#72227255

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#93269
commutator	C₄×C₁₆	C₈²
Frattini, cosocle	#72216182	C₂²
socle	C₂²	#72216542
prime radical	C₂³	#92938
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₁₆ ◁ C₈×#346 ◁ #72216182 ◁ G
LECS	G ▷ #72216182 ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^12|a^2=c,b^2=dehjl,c^2=f,d^2=gj,e^2=h,f^2=l,g^2=i,h^2=j,i^2=k,j^2=l,b^a=bd,c^b=cijkl,d^a=dg,d^c=dk,e^a=ejl,e^c=el,f^b=fkl,g^a=gi,h^a=hl,i^a=ik

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227257

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