#72341063

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

2388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=gi,d^2=fgi,e^2=fhi,b^a=bi,c^a=cgj,c^b=cgj,d^a=dgi,d^b=dgh,d^c=df,e^a=eghi,e^b=efghj,e^c=ehij,e^d=efj,f^a=fj,f^b=fi,f^c=fj,f^d=fj,f^e=fi,g^d=gj,g^e=gij,h^b=hj,h^c=hij,h^d=hij,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341113
maximal normal subgroup of	#72341083, #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.9);;
gap> SetPower(coll, 5, F.6*F.8*F.9);;
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.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*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.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
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.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);