#72258170

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#128572
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72258169
prime radical	C₂×#508	#97
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ #72225138 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1138 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:

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

Polycyclic presentation:

2^12|a^2=l,b^2=d,c^2=e,d^2=g,e^2=h,f^2=i,h^2=j,i^2=k,k^2=l,b^a=bf,c^a=cf,c^b=cfhkl,d^a=dkl,d^c=dj,e^a=ei,e^b=eijl,f^a=fikl,f^b=fil,f^d=fl,g^a=gl,h^a=hk,h^b=hk,i^a=ikl,i^b=ik,j^a=jl,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258194
maximal normal subgroup of	#72258194

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