#72286703

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72284771
commutator	#3151	C₂²×C₄
Frattini, cosocle	C₂×#3151	C₂³
prime radical	#6348452	C₂²
LCS	G ▷ #3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂×#3151 ◁ G
LECS	G ▷ C₂×#3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2169 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=gijk,c^2=k,d^2=k,e^2=k,h^2=jk,i^2=k,b^a=bg,c^a=ce,c^b=cf,d^b=djk,e^a=ek,e^b=eh,e^c=ek,f^b=fhj,g^a=gjk,g^b=gjk,g^c=ghj,g^d=gk,g^e=gij,g^f=gijk,h^a=hi,h^b=hik,h^c=hjk,h^e=hjk,h^f=hjk,h^g=hjk,i^a=ik,i^b=ij,i^c=ijk,i^e=ik,i^f=ik,i^g=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286925
maximal normal subgroup of	#72286925

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