#72342965

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#130798
commutator	C₂²×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×#504	C₂³
prime radical	#7623826	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #33433 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=efhj,e^2=hij,f^2=hij,h^2=ij,i^2=j,b^a=bg,c^a=ce,c^b=cd,d^c=dg,e^a=eh,e^b=ej,e^c=ehj,f^a=fh,f^b=fgj,f^c=fhj,f^d=fj,g^c=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342966, #72342984, #72343025
maximal normal subgroup of	#72342966, #72342984, #72343025

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