#72258137

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#128688
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂³
socle	C₂	#72258136
prime radical	C₂×#508	C₂×C₈
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

543 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=f,c^2=d,d^2=g,e^2=hk,f^2=ijk,g^2=i,h^2=j,j^2=k,b^a=be,c^a=ceh,c^b=ceij,d^a=dh,d^b=dhjk,e^a=ehj,e^b=ehj,e^c=ejk,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

164 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.11);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*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.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9*F.10);;
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*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
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);