#72283166

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#246503
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#7563313	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=e,d^2=j,e^2=j,f^2=h,g^2=i,h^2=j,b^a=bi,c^a=ce,c^b=cf,d^a=dj,d^b=dg,e^a=ej,e^b=ej,f^b=fj,f^c=fh,f^d=fh,g^c=gh,g^d=gi,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283316, #72283626
maximal normal subgroup of	#72283316, #72283626

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