#72342801

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	#8541
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#106132
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

3731 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=hi,d^2=j,e^2=j,b^a=bg,c^a=ce,c^b=cehj,d^b=dh,d^c=dhi,e^a=ehij,e^b=ehi,e^c=ej,f^a=fh,f^c=fhij,f^e=fj,g^a=gh,g^b=gh,g^c=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342803, #72342838
maximal normal subgroup of	#72342803, #72342838

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