#72287443

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#12067
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
prime radical	#72234321	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #72231883 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^12|b^2=jl,c^2=f,d^2=jl,e^2=fh,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bjl,c^b=cjl,d^a=djl,d^c=dg,e^d=eikl,f^d=fkl,g^c=gi,g^d=gikl,g^e=gk,g^f=gl,h^d=hl,i^c=ik,i^d=ikl,i^e=il,k^c=kl,k^d=kl

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

Relationships

automorphism group of

#7541, #7542, #13742, …

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