#72287718

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,c^2=f,e^2=fh,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=ci,c^b=chj,d^a=dj,d^c=dhj,e^a=egij,e^b=ehj,e^d=ej,f^a=fj,g^c=gj,g^e=gi,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287726, #72287751, #72319259
maximal normal subgroup of	#72287726, #72287751, #72319259

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