#72225494

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#140433
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂³	#12447
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|a^2=e,b^2=ehjk,c^2=fhj,d^2=gij,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bd,c^a=cgj,c^b=cf,d^a=dg,d^b=dgjk,e^b=ej,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549, #72227135, #72227154
maximal normal subgroup of	#72225549, #72227135, #72227154
maximal quotient of	#72225495

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