#72281480

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#116855
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂×#585	D₄
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2306 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dg,c^2=di,e^2=i,f^2=ij,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cf,d^c=dj,e^a=ei,e^b=eij,e^c=ei,f^a=fh,f^b=fhj,f^c=fij,g^c=ghij,g^f=gj,h^a=hj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281673, #72282517
maximal normal subgroup of	#72281673, #72282517

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