#72287874

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	#7623516	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2208 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)(4,10)(5,11)(6,12)(7,9)(13,23)(14,21)(15,22)(16,25)(17,26)(18,27)(19,28)(20,24)(29,31)(30,32)(34,39)(40,54)(41,55)(42,53)(44,59)(49,64)(57,63)(58,62);(1,3)(2,8)(4,16)(5,18)(6,14)(7,13)(9,23)(10,25)(11,27)(12,21)(15,20)(17,29)(19,30)(22,24)(26,31)(28,32)(35,43)(40,54)(44,59)(46,60)(47,56)(48,61)(57,63)(58,62);(2,9)(3,13)(10,12)(11,24)(14,16)(15,18)(26,28)(29,30)(33,34,38,39)(35,44,43,59)(36,49,52,64)(37,53,51,42)(40,56,54,47)(41,50,55,45)(46,62,60,58)(48,57,61,63);(1,4,7,6)(2,10,9,12)(3,14,13,16)(5,19,20,17)(8,21,23,25)(11,28,24,26)(15,30,18,29)(22,32,27,31)(33,35,45,46,37,48,52,56,38,43,50,60,51,61,36,47)(34,40,49,63,42,58,55,59,39,54,64,57,53,62,41,44);(1,5,6,19,7,20,4,17)(2,11,12,28,9,24,10,26)(3,15,14,29,13,18,16,30)(8,22,21,31,23,27,25,32)(34,39)(35,46,43,60)(36,50)(37,51)(40,57,54,63)(41,49)(44,58,59,62)(45,52)(47,61,56,48)(55,64)

Polycyclic presentation:

2^10|c^2=j,d^2=ghi,e^2=f,f^2=h,g^2=ij,i^2=j,c^a=chj,c^b=ch,d^b=dhj,d^c=dg,e^b=eh,e^c=ej,e^d=egh,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318539, #72319199, #72319381
maximal normal subgroup of	#72318539, #72319199, #72319381

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