#72258141

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#129165
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂³
socle	C₂	#72258139
prime radical	C₂³	#12068
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ #149502 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^11|a^2=f,b^2=fk,c^2=d,d^2=g,e^2=hjk,f^2=ijk,g^2=i,h^2=j,j^2=k,b^a=be,c^a=cei,c^b=cehi,d^a=dh,d^b=dhjk,e^a=eh,e^b=eh,e^c=ej,e^d=ek,f^c=fk,g^a=gj,g^b=gjk,h^a=hjk,h^b=hjk,h^c=hk,i^a=ik,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258148
maximal normal subgroup of	#72258148

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