#72319952

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#368434
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#60450	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2030 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)(3,16)(4,11)(5,9)(6,15)(7,14)(8,12)(10,21)(13,28)(17,46)(18,50)(19,38)(20,36)(22,37)(23,45)(24,32)(25,44)(26,40)(27,34)(29,33)(30,42)(31,49)(35,52)(39,53)(41,57)(43,48)(47,64)(51,59)(54,63)(55,60)(56,62)(58,61);(2,9,12,33)(3,17,19,48)(5,27)(6,7)(11,32,40,61)(13,14,34,15)(16,38)(18,47,23,54)(20,22,49,21)(24,56)(25,30)(28,29)(31,46)(36,43)(41,44,62,42)(45,50)(51,52,55,53)(57,58)(59,63)(60,64);(1,3,6,22,8,19,7,21)(2,10,15,16,12,37,14,38)(4,23,30,53,26,18,25,52)(5,17,27,20,29,48,28,49)(9,31,34,46,33,36,13,43)(11,39,42,50,40,35,44,45)(24,54,56,55,58,47,57,51)(32,60,62,64,61,59,41,63);(1,4,8,26)(2,11,12,40)(3,18,19,23)(5,24,29,58)(6,30,7,25)(9,32,33,61)(10,35,37,39)(13,41,34,62)(14,44,15,42)(16,45,38,50)(17,47,48,54)(20,51,49,55)(21,53,22,52)(27,56,28,57)(31,59,36,60)(43,64,46,63)

Polycyclic presentation:

2^10|b^2=e,c^2=eij,d^2=j,f^2=ij,g^2=j,i^2=j,b^a=bf,c^a=cg,c^b=cfij,d^a=dh,d^c=dj,e^a=ej,e^c=ej,f^a=fi,f^b=fij,f^c=fij,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72320114, #72343936
maximal normal subgroup of	#72320043, #72320114, #72343936

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