#72303805

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10736
commutator	C₂³×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#123862
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#568 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3755 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)(3,13)(4,9,20,34)(5,11)(6,12)(7,10)(8,16)(14,44,46,42)(15,39)(17,43)(18,50,49,63)(19,52)(21,35,54,62)(22,36,27,41)(23,31)(24,33)(25,37)(26,58)(28,38)(29,47,59,56)(30,48)(32,55,60,51)(40,45)(53,64,57,61);(1,3)(2,8)(4,18)(5,23,25,16)(6,26,28,17)(7,15)(9,32)(10,31)(11,39,37,13)(12,40,38,30)(14,27,46,22)(19,45)(20,49)(21,55,54,51)(24,48)(29,36,59,41)(33,43)(34,60)(35,63,62,50)(42,64)(44,61)(47,57)(52,58)(53,56);(1,4,19,53,7,20,24,57)(2,9,33,61,10,34,52,64)(3,14,45,51,15,46,48,55)(5,21,28,22,25,54,6,27)(8,29,43,50,31,59,58,63)(11,35,12,41,37,62,38,36)(13,42,30,32,39,44,40,60)(16,47,26,18,23,56,17,49)

Polycyclic presentation:

2^10|a^2=d,b^2=h,c^2=defgij,g^2=j,b^a=bf,c^a=cde,c^b=cegj,d^c=dj,e^a=ej,e^b=ej,f^b=fi,g^b=gi,g^c=gj,h^a=hi,h^c=hi,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303815, #72312278, #72312767
maximal normal subgroup of	#72303815, #72312278, #72312767

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