#72225097

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#123403
commutator, Frattini, cosocle	C₁₆²	C₂³
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|c^2=eg,d^2=fh,e^2=gi,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=ce,c^b=ce,d^a=df,d^b=df,e^a=eg,e^b=eg,f^a=fh,f^b=fh,g^a=gi,g^b=gi,h^a=hj,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

518 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225108, #72227076
maximal normal subgroup of	#72225108, #72227076

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