#72250043

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#185645
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10576944	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3671 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=hj,d^2=efgij,e^2=j,f^2=i,g^2=j,i^2=j,d^a=de,d^b=df,d^c=di,e^a=ej,e^d=eh,f^b=fi,f^c=fj,g^a=gh,g^b=gj,g^d=gh,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237, #72281944
maximal normal subgroup of	#72250237, #72281944
maximal quotient of	#72250440

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