#72225446

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#1619
commutator	C₄×C₈	C₂²×C₈
Frattini, cosocle	C₄²×C₈	C₂³
socle	C₂	#99507
prime radical	#33594	C₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=d,d^2=g,e^2=hj,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bfj,c^a=ce,c^b=cfi,d^a=dh,d^b=di,e^a=ehj,e^b=ej,e^c=ej,f^b=fij,f^c=fj,g^a=gj,g^b=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225447, #72225483, #72225498
maximal normal subgroup of	#72225447, #72225483, #72225498

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