#72343434

order	2048
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	#72285170
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#72320934	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #479961 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=hk,d^2=gijk,e^2=h,f^2=ik,g^2=j,i^2=j,j^2=k,b^a=bik,c^a=cf,c^b=cf,d^a=dfhij,d^b=dfhik,d^c=dg,e^b=ek,e^c=ej,e^d=ejk,f^a=fi,f^b=fijk,f^c=fij,f^d=fhij,f^e=fk,g^a=ghjk,g^b=ghj,g^c=gjk,g^d=gjk,g^f=gk,h^c=hk,h^d=hk,i^a=ij,i^b=ijk,i^c=ijk,i^d=ijk,j^a=jk,j^b=jk,j^c=jk,j^d=jk

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343465
maximal normal subgroup of	#72343465

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