#72284707

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

2299 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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=f,d^2=hi,e^2=i,b^a=bi,c^a=ci,c^b=ch,d^a=dhj,d^b=di,d^c=dg,e^a=efi,e^b=egj,e^d=ef,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285517
maximal normal subgroup of	#72285517

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