#72340298

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72213808	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20586 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=ik,b^2=ghijk,c^2=j,d^2=k,e^2=hj,f^2=k,b^a=bf,c^a=ce,d^b=di,d^c=dhjk,e^a=eh,e^b=eg,e^c=ehjk,f^b=fik,f^c=fgi,f^e=fijk,g^a=gi,g^b=gjk,g^c=gj,g^d=gk,g^e=gj,h^a=hk,h^b=hj,h^f=hk,i^b=ik,i^c=ij,i^e=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340332, #72340478, #72340497
maximal normal subgroup of	#72340332, #72340478, #72340497

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