#72343148

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10948
commutator	C₂×M₁₆	C₂×C₄²
Frattini, cosocle	C₄×#131	C₂³
socle	C₂	#111118
prime radical	#3584	C₂×C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=j,c^2=egi,d^2=h,e^2=i,f^2=j,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=cfi,c^b=ce,d^b=dj,e^a=ehij,e^b=eij,e^c=eij,f^b=fj,f^c=fhij,f^e=fj,g^a=gj,g^b=gij,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343157, #72343163, #72343170, …
maximal normal subgroup of	#72343157, #72343163, #72343170, …

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