#72225105

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

Aliases

No aliases for this group.

Subgroups

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

555 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:

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

Polycyclic presentation:

2^11|b^2=j,c^2=egk,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, #72227082
maximal normal subgroup of	#72225108, #72227082

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.10);;
gap> SetPower(coll, 3, F.5*F.7*F.11);;
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);