#72287487

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	#7623514	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2215 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)(3,9)(4,19)(5,21)(6,12)(7,10)(8,13)(11,22)(14,30)(15,32)(16,26)(17,24)(18,27)(20,28)(23,29)(25,31)(33,34)(35,45)(36,49)(37,43)(38,44)(39,41)(40,48)(42,53)(46,56)(47,57)(50,60)(51,61)(52,62)(54,63)(55,59)(58,64);(1,3)(2,9)(4,14)(5,15)(6,23)(7,17)(8,18)(10,24)(11,25)(12,29)(13,27)(16,20)(19,30)(21,32)(22,31)(26,28);(1,4,8,7)(2,10,13,19)(3,14,18,17)(5,20,22,6)(9,24,27,30)(11,28,21,12)(15,16,31,23)(25,26,32,29)(33,35,37,46,38,36,50,54,39,48,47,64,55,53,52,51)(34,40,43,58,44,42,60,61,41,45,57,56,59,49,62,63);(1,5,7,6,8,22,4,20)(2,11,10,28,13,21,19,12)(3,15,17,23,18,31,14,16)(9,25,24,26,27,32,30,29)(34,41)(35,46,48,64)(36,51,53,54)(37,50)(38,55)(40,56,45,58)(42,61,49,63)(43,62)(47,52)(57,60);(1,6,7,22,8,20,4,5)(2,12,10,11,13,28,19,21)(3,16,17,15,18,23,14,31)(9,26,24,32,27,29,30,25)(33,36,52,64,55,35,47,54,39,53,50,46,38,48,37,51)(34,42,62,56,59,40,57,61,41,49,60,58,44,45,43,63)

Polycyclic presentation:

2^10|c^2=g,d^2=f,e^2=fghij,f^2=h,g^2=i,i^2=j,c^a=chj,d^a=dhj,d^c=dghij,e^a=ej,e^b=eh,e^c=eij,e^d=egj,f^c=fj,f^e=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72287829, #72318539
maximal normal subgroup of	#72287517, #72287829, #72318539

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