#72283662

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	#72213804	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

3507 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)(3,8)(4,17,21,37)(5,10)(6,11)(7,12)(9,20,34,53)(13,40,42,64)(14,28)(15,36)(16,30)(18,47,43,62)(19,48,22,54)(23,56)(24,38)(25,32)(26,39)(27,41,58,45)(29,46)(31,51,63,49)(33,55,61,50)(35,52)(44,59)(57,60);(1,3)(2,8)(4,18)(5,14,25,23)(6,15)(7,16)(9,31,34,63)(10,28,32,56)(11,36)(12,30)(13,19)(17,47)(20,51,53,49)(21,43)(22,42)(24,44,52,57)(26,46)(27,33,58,61)(29,39)(35,60,38,59)(37,62)(40,48)(41,55,45,50)(54,64);(1,4,5,22,7,21,25,19)(2,9,32,61)(3,13,23,18)(6,20,52,55)(8,27,28,31,30,58,56,63)(10,33,12,34)(11,37,38,48,39,17,35,54)(14,43,16,42)(15,45,44,51,46,41,57,49)(24,50,26,53)(29,40,59,47)(36,64,60,62);(1,3)(2,8)(4,13)(5,23)(6,15)(7,16)(9,27)(10,28)(11,36)(12,30)(14,25)(17,40)(18,19)(20,45)(21,42)(22,43)(24,44)(26,46)(29,39)(31,61)(32,56)(33,63)(34,58)(35,59)(37,64)(38,60)(41,53)(47,48)(49,50)(51,55)(52,57)(54,62)

Polycyclic presentation:

2^11|a^2=jk,b^2=gj,c^2=eghik,e^2=hik,f^2=hk,c^a=cf,c^b=ce,d^a=dgj,d^b=dgj,d^c=dgijk,e^a=eg,e^b=eh,e^c=eik,e^d=eij,f^a=fh,f^b=fgik,f^c=fgh,f^d=fijk,f^e=fik,g^c=gi,g^e=gj,g^f=gjk,h^b=hj,h^c=hjk,h^d=hk,h^e=hk,i^a=ij,i^b=ijk,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285547, #72340399, #72340497
maximal normal subgroup of	#72285547, #72340399, #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.10*F.11);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.5*F.7*F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.8*F.9*F.11);;
gap> SetPower(coll, 6, F.8*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, 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);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);