#72225327

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#12067
commutator	C₁₆	C₂²×C₄×C₈
Frattini, cosocle	C₂×#346	C₂⁴
socle	C₂	C₄×#12067
prime radical	#33981	C₂×C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#1313 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1401 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:

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

Polycyclic presentation:

2^11|a^2=k,c^2=e,d^2=g,e^2=h,f^2=ij,i^2=jk,j^2=k,b^a=bf,c^a=ci,d^a=djk,d^b=djk,e^a=ej,f^a=fi,f^b=fi,f^c=fj,f^e=fk,g^a=gk,g^b=gk,h^a=hk,i^a=ij,i^b=ij,i^c=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227138, #72227158
maximal normal subgroup of	#72225336, #72227138, #72227158

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