#72287430

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	#7623529	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17316 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2216 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)(5,18)(6,10)(7,12)(11,15)(13,21)(14,23)(16,26)(17,27)(19,22)(20,24)(25,30)(28,32)(29,31)(33,34)(35,40)(36,48)(37,43)(38,44)(39,41)(42,50)(45,62)(46,57)(47,54)(49,59)(51,61)(52,60)(53,56)(55,63)(58,64);(1,3,6,14,7,15,5,19)(2,8,10,23,12,11,18,22)(4,13,16,29,17,30,20,28)(9,21,26,31,27,25,24,32)(33,35,37,38,39,47,53,46)(34,40,43,44,41,54,56,57)(36,49,51,45,50,64,52,63)(42,58,60,55,48,59,61,62);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,21)(10,26)(11,25)(12,27)(14,29)(15,30)(18,24)(19,28)(22,32)(23,31)(34,41)(36,50)(40,54)(43,56)(44,57)(45,63)(49,64)(51,52);(1,5,7,6)(2,10,12,18)(3,14,15,19)(4,16,17,20)(8,22,11,23)(9,24,27,26)(13,28,30,29)(21,31,25,32)(33,36,38,49,37,52,47,63,39,50,46,64,53,51,35,45)(34,42,44,58,43,61,54,62,41,48,57,59,56,60,40,55);(1,3,5,19,7,15,6,14)(2,11,18,23,12,8,10,22)(4,13,20,28,17,30,16,29)(9,25,24,31,27,21,26,32)(34,41)(35,46)(36,49,50,64)(37,53)(38,47)(40,44)(42,59,48,58)(45,51,63,52)(54,57)(55,61,62,60)

Polycyclic presentation:

2^10|b^2=f,d^2=g,e^2=fhi,f^2=h,g^2=i,i^2=j,c^a=cj,d^a=dh,d^b=dhij,d^c=dh,e^a=eh,e^b=ehij,e^d=egij,f^d=fj,f^e=fj,g^b=gj,g^e=gi,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287479, #72287827, #72287851
maximal normal subgroup of	#72287479, #72287827, #72287851

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