#72257883

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#7867
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₄×#97	C₂³
socle	C₂	#95040
prime radical	C₂⁴	#344
LCS	G ▷ C₂×C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #20331 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

719 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)(3,8,16,29,19,31,18,39)(4,9)(5,24,14,11)(6,10)(7,13)(12,25)(15,32,44,58,20,28,43,54)(17,30,47,56,40,38,46,52)(21,36)(22,50,41,34)(23,35)(26,27,42,37)(33,49)(45,59,61,63,48,55,62,64)(51,53,60,57);(1,3,14,40,6,16,42,47,7,19,5,17,25,18,26,46)(2,8,27,52,12,39,24,30,13,31,37,56,10,29,11,38)(4,20,22,48,23,43,51,61,21,15,41,45,49,44,60,62)(9,32,57,63,33,54,34,55,36,28,53,64,35,58,50,59);(1,4)(2,9)(3,15)(5,22)(6,23)(7,21)(8,28)(10,35)(11,34)(12,33)(13,36)(14,41)(16,44)(17,45)(18,43)(19,20)(24,50)(25,49)(26,51)(27,53)(29,54)(30,55)(31,32)(37,57)(38,59)(39,58)(40,48)(42,60)(46,62)(47,61)(52,63)(56,64)

Polycyclic presentation:

2^10|a^2=e,b^2=defhj,d^2=gij,e^2=h,f^2=ij,g^2=j,i^2=j,b^a=bd,c^a=ch,c^b=ch,d^a=dg,d^b=dg,d^c=dj,e^b=eij,f^a=fgij,f^b=fgi,f^e=fj,g^a=gj,g^b=gj,h^b=hj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265296, #72265537
maximal normal subgroup of	#72265296, #72265537

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