#72322953

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	C₂×C₄	C₂×#12067
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
socle	C₂²	C₂²×#12067
prime radical	#72322435	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#7053 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^12|a^2=f,c^2=fh,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bg,c^b=cikl,d^b=dj,e^b=el,f^b=fkl,g^a=gi,g^b=gikl,g^c=gk,g^f=gl,h^b=hl,i^a=ik,i^b=ikl,i^c=il,k^a=kl,k^b=kl

Conjugacy classes & character table

280 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, 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.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, 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, 7, 2, F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*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);