#72225360

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₂²×D₁₆
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂⁴
socle	C₂	C₂×C₁₆×D₁₆
prime radical	#33981	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

863 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)(3,16)(4,8)(5,11)(6,13)(7,10)(9,19)(12,17)(14,42)(15,27)(18,45)(20,28)(21,43)(22,30)(23,31)(24,29)(25,40)(26,38)(32,56)(33,52)(34,53)(35,47)(36,49)(37,61)(39,48)(41,55)(44,62)(46,51)(50,58)(54,57)(59,63)(60,64);(2,8,11,30,13,31,40,39,10,29,38,41,15,32,14,12)(9,28,33,43,34,57,59,45,36,58,60,46,37,44,35,16);(1,3,17,47,42,62,56,61,27,51,55,64,26,50,24,49,7,18,48,63,25,54,23,53,6,21,22,52,5,20,4,19)(2,9,8,28,11,33,30,43,13,34,31,57,40,59,39,45,10,36,29,58,38,60,41,46,15,37,32,44,14,35,12,16);(2,10)(3,18)(8,29)(11,38)(12,39)(13,15)(14,40)(19,49)(20,50)(21,51)(30,41)(31,32)(47,63)(52,64)(53,61)(54,62)

Polycyclic presentation:

2^11|b^2=e,c^2=fhjk,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cf,d^a=dk,d^c=dk,e^a=eh,f^a=fhjk,g^a=gj,h^a=hjk,i^a=ik,j^a=jk

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225393, #72225451, #72225464, …
maximal normal subgroup of	#72225393, #72225451, #72225464, …

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