#72341070

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

2306 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)(4,18)(5,10)(6,23)(7,12)(8,17)(9,20)(11,19)(14,25)(15,29)(16,27)(21,26)(22,28)(24,30)(31,32)(33,34)(35,43)(36,40)(37,50)(38,42)(39,46)(41,53)(44,54)(45,55)(47,60)(48,59)(49,57)(51,58)(52,56)(61,64)(62,63);(1,3)(2,8)(4,14,20,32)(5,21)(6,24,19,16)(7,17)(9,25,18,31)(10,29)(11,30,23,27)(12,13)(15,22)(26,28)(33,35)(34,39)(36,47)(37,51,53,45)(38,46)(40,54)(41,58,50,55)(42,43)(44,49)(48,62,52,61)(56,63,59,64)(57,60);(1,4)(2,9)(3,14,17,32)(5,6,22,19)(7,20)(8,25,13,31)(10,23,28,11)(12,18)(15,16)(21,24)(26,30)(27,29)(33,36)(34,40,42,57)(35,44,46,47)(37,52,53,48)(38,49)(39,54)(41,59)(43,60)(45,61)(50,56)(51,62)(55,63,58,64);(1,5,7,22)(2,10,12,28)(3,15)(4,19)(6,20)(8,26)(9,11)(13,29)(14,24,32,16)(17,21)(18,23)(25,27,31,30)(33,37,38,53)(34,41)(35,45,46,51)(36,48)(39,55)(40,56,57,59)(42,50)(43,58)(44,61)(47,62)(49,52)(54,63,60,64);(1,6,7,19)(2,11,12,23)(3,16)(4,5)(8,27)(9,28)(10,18)(13,30)(14,21,32,15)(17,24)(20,22)(25,26,31,29)(33,36)(34,40,42,57)(35,44,46,47)(37,48)(38,49)(39,54)(41,59,50,56)(43,60)(45,62,51,61)(52,53)(55,63)(58,64)

Polycyclic presentation:

2^10|b^2=g,c^2=fghij,d^2=fij,e^2=fhj,b^a=bij,c^a=ch,c^b=cf,d^a=dg,d^b=dhij,d^c=dgh,e^a=egh,e^b=efi,e^c=egi,e^d=efgh,f^a=fj,f^b=fi,f^c=fij,f^e=fij,g^c=gi,g^d=gj,g^e=gi,h^b=hj,h^c=hj,h^d=hi,h^e=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341110, #72341113
maximal normal subgroup of	#72341083, #72341110, #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.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8);;
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.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);