#72286122

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	#72237744
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72284401	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

2147 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)(3,9)(4,18)(5,11)(6,12)(7,13)(8,14)(10,22)(15,47)(16,34)(17,36)(19,39)(20,38)(21,58)(23,59)(24,54)(25,37)(26,43)(27,35)(28,44)(29,45)(30,53)(31,33)(32,49)(40,48)(41,55)(42,56)(46,57)(50,63)(51,62)(52,64)(60,61);(1,3)(2,9)(4,19,24,56)(5,7)(6,16,29,52)(8,17)(10,37,41,58)(11,33)(12,34,45,64)(13,35)(14,36)(15,48,50,57)(18,39)(20,26,28,60)(21,55)(22,25)(23,49)(27,31)(30,51)(32,59,62,53)(38,61,44,43)(40,47)(42,54)(46,63);(1,4,20,57)(2,10,38,53)(3,15,26,56,17,50,60,19)(5,25,16,51)(6,22,31,23,29,55,7,30)(8,24,28,48)(9,32,61,58,36,62,43,37)(11,42,64,47)(12,18,33,40,45,54,13,46)(14,41,44,59)(21,52,49,27)(34,63,35,39);(1,5,8,27)(2,11,14,35)(3,7)(4,21,24,25)(6,26)(9,33)(10,39,41,42)(12,43)(13,36)(15,30)(16,28,52,20)(17,31)(18,37)(19,55)(22,56)(23,50)(29,60)(32,40)(34,38,64,44)(45,61)(46,62)(47,53,63,59)(48,51,57,49)(54,58)

Polycyclic presentation:

2^11|b^2=ghij,c^2=efgijk,d^2=hij,f^2=hj,b^a=bijk,c^a=ce,c^b=cefhj,d^a=dgjk,d^b=dij,d^c=dhij,e^b=eg,e^c=egij,e^d=eij,f^a=fgij,f^b=fgh,f^c=fgj,f^d=fi,f^e=fij,g^b=gk,g^c=gi,g^d=gj,g^f=gjk,h^a=hj,h^b=hjk,h^c=hj,h^d=hk,h^e=hk,i^b=ij,i^c=ijk,i^d=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286183
maximal normal subgroup of	#72286183

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