#72287635

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72287578
commutator	C₂²×C₁₆	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	#3444	SD₁₆
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622660 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1467 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=efijk,c^2=d,d^2=h,e^2=k,f^2=i,g^2=j,i^2=j,j^2=k,b^a=be,c^b=cefgij,d^b=dj,e^b=eh,e^c=ek,f^a=fhk,f^b=fh,f^c=fi,f^d=fk,f^e=fk,g^a=gh,g^b=gk,g^c=gjk,g^e=gk,g^f=gk,h^b=hk,i^c=ij,j^c=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287845, #72343364, #72343482
maximal normal subgroup of	#72287845, #72343364, #72343482

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