#72281038

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10948
commutator	C₂×M₁₆	C₂×C₄²
Frattini, cosocle	C₂²×#131	C₂³
socle	C₂	#111110
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2320 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=d,c^2=deij,e^2=i,f^2=i,i^2=j,b^a=bg,c^a=cf,c^b=cfhij,d^c=dh,e^a=ei,e^b=eij,e^c=eh,f^a=fij,f^b=fhi,f^c=fj,f^d=fj,f^e=fj,g^c=ghj,g^f=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281230, #72282517
maximal normal subgroup of	#72281230, #72282517

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