#72341065

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gi,d^2=fgh,e^2=fg,b^a=bi,c^a=cgj,c^b=cgj,d^a=dh,d^b=df,d^c=dfh,e^a=egh,e^b=efghi,e^c=egh,e^d=efhij,f^a=fj,f^b=fi,f^c=fj,f^d=fij,f^e=fij,g^d=gi,g^e=gij,h^b=hj,h^c=hij,h^d=hj,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341091, #72341113
maximal normal subgroup of	#72341083, #72341091, #72341113

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