#72341082

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

2297 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)(3,11)(4,15,18,8)(5,19)(6,10)(7,14)(9,22)(12,25,16,23)(13,24)(17,28,21,26)(20,27)(29,32,30,31)(33,34)(35,39)(36,46,49,40)(37,50,52,41)(38,42)(43,56,47,53)(44,54,61,58)(45,55)(48,59)(51,57)(60,63)(62,64);(1,3)(2,7)(4,12,18,16)(5,20)(6,14)(8,25,15,23)(9,27)(10,11)(13,22)(17,30,21,29)(19,24)(26,31,28,32)(33,35,38,45)(34,39,42,55)(36,47)(37,44)(40,56)(41,58)(43,49)(46,53)(48,62,51,60)(50,54)(52,61)(57,63,59,64);(1,3)(2,7,10,11)(4,16)(5,13,22,20)(6,14)(8,23,15,25)(9,24)(12,18)(17,30,21,29)(19,27)(26,31)(28,32)(33,36,38,49)(34,40,42,46)(35,43)(37,51,52,48)(39,53)(41,59,50,57)(44,60)(45,47)(54,63)(55,56)(58,64)(61,62);(1,4,6,18)(2,8)(3,12,14,16)(5,21)(7,23)(9,28,19,26)(10,15)(11,25)(13,29)(17,22)(20,30)(24,31,27,32)(33,37)(34,41)(35,44,45,61)(36,48)(38,52)(39,54,55,58)(40,57)(42,50)(43,60,47,62)(46,59)(49,51)(53,63,56,64);(1,5,6,22)(2,9,10,19)(3,13,14,20)(4,17,18,21)(7,24,11,27)(8,26,15,28)(12,29,16,30)(23,31,25,32)(36,49)(41,50)(43,47)(54,58)

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=fg,d^2=fgij,e^2=i,b^a=bi,c^a=cgh,c^b=cf,d^a=dfghij,d^b=dfhij,d^c=dfgi,e^a=egi,e^b=ei,e^c=eghij,e^d=eh,f^a=fi,f^b=fj,f^c=fj,f^d=fij,f^e=fi,g^c=gj,g^d=gij,h^a=hj,h^b=hi,h^c=hij,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341083, #72341103, #72341113
maximal normal subgroup of	#72341083, #72341103, #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, 1, F.7);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
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);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);