#72226958

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226957
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	C₂×#72214187	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|c^2=e,d^2=fh,e^2=ik,f^2=hj,g^2=il,h^2=j,i^2=k,k^2=l,c^a=cg,c^b=cgjkl,d^a=dgkl,d^b=df,d^c=dj,e^a=ei,e^b=eil,f^a=fikl,f^b=fh,g^a=gik,g^b=gik,g^c=gl,h^a=hk,h^b=hj,i^a=ikl,i^b=ikl,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982
maximal normal subgroup of	#72226982

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