#72225417

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#129096
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×M₆₄	C₂³
socle	C₂	#72225415
prime radical	C₂³	#151045
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1397 ◁ #149495 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^12|a^2=d,b^2=e,c^2=fghikl,d^2=g,f^2=hj,g^2=i,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bhijkl,c^a=cf,c^b=cjk,d^b=dk,e^a=ejkl,e^d=el,f^a=fh,f^b=fl,g^b=gl,h^a=hj,j^a=jl

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550, #72227029, #72227059
maximal normal subgroup of	#72225550, #72227029, #72227059

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