#72280720

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	#10918
commutator	C₂×#41	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#124038
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3927 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)(4,17,21,44)(5,10)(6,12)(7,11)(9,20,32,48)(13,39,41,61)(14,28)(15,37)(16,29)(18,43,47,58)(19,36)(22,45,52,63)(23,34)(24,35)(25,38)(26,30)(27,40,55,53)(31,50,59,46)(33,51,60,64)(42,57)(49,62)(54,56);(1,3)(2,8)(4,18)(5,23)(6,16)(7,26,25,15)(9,31)(10,34)(11,37,38,30)(12,29)(13,22,41,52)(14,19)(17,43)(20,50)(21,47)(24,42,49,54)(27,33,55,60)(28,36)(32,59)(35,56,62,57)(39,63,61,45)(40,64,53,51)(44,58)(46,48);(1,4,19,48,6,21,5,20)(2,9,10,17,12,32,36,44)(3,13,14,40,16,41,23,53)(7,22,49,64,25,52,24,51)(8,27,34,61,29,55,28,39)(11,33,35,45,38,60,62,63)(15,18,42,46,26,47,54,50)(30,31,57,43,37,59,56,58)

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=de,d^2=j,b^a=bh,c^a=cd,c^b=cg,d^b=dj,e^a=ej,e^b=ej,f^c=fj,g^a=gi,g^b=gi,h^c=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280762, #72281160, #72281807
maximal normal subgroup of	#72280762, #72281160, #72281807

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