#72320904

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#137652
commutator	C₂²×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×#346	C₂³
prime radical	#33617	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #33326 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=j,c^2=i,d^2=g,e^2=h,h^2=i,i^2=j,b^a=be,c^a=cf,c^b=cfgj,d^b=dij,e^a=eh,e^b=ehij,e^c=ej,e^d=ej,f^b=fj,f^c=fj,g^b=gj,h^a=hi,h^b=hij,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320965, #72320986, #72321237
maximal normal subgroup of	#72320965, #72320986, #72321237

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