#72342836

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9495
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#110401
prime radical	#62811	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2690 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=i,d^2=j,b^a=bg,c^a=ce,c^b=cegh,d^b=dh,d^c=di,e^a=ei,e^b=ehi,f^a=fh,f^c=fij,f^e=fj,g^a=gh,g^b=gh,g^c=gj,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342838, #72342852, #72342855
maximal normal subgroup of	#72342838, #72342852, #72342855

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