#72343278

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	#72343249
commutator	C₂²×C₁₆	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	C₂×#3415	C₂×C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #479960 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1624 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=egij,d^2=h,e^2=ik,f^2=ik,g^2=ik,i^2=j,j^2=k,b^a=bhi,c^a=ce,c^b=cf,d^b=dk,d^c=dj,e^a=ei,e^b=eij,e^c=ehijk,e^d=ek,f^a=fi,f^b=fij,f^c=fij,f^d=fk,g^a=ghik,g^b=gij,g^c=ghijk,g^d=gk,g^e=gk,h^c=hk,i^a=ij,i^b=ijk,i^c=ijk,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343292, #72343385, #72343423
maximal normal subgroup of	#72343292, #72343385, #72343423

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