#72342821

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	#9495
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#106542
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

3050 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)(3,13,15,39,17,40,44,64)(4,9,19,31,21,33,49,59)(6,11,22,34,25,37,51,61)(8,26,28,45,29,53,56,16)(14,38,41,62,18,46,43,63)(20,36,48,60,24,32,50,58)(27,52,54,42,30,57,55,47);(1,3,6,16)(2,8,11,13)(4,18,24,47)(5,15,22,26)(7,17,25,45)(9,30,36,46)(10,28,34,39)(12,29,37,40)(14,20,42,21)(19,41,50,57)(23,44,51,53)(27,32,38,33)(31,54,60,62)(35,56,61,64)(43,48,52,49)(55,58,63,59);(1,4)(2,9)(3,14)(5,19)(6,24)(7,21)(8,27)(10,31)(11,36)(12,33)(13,38)(15,43)(16,42)(17,18)(20,25)(22,50)(23,49)(26,52)(28,55)(29,30)(32,37)(34,60)(35,59)(39,63)(40,46)(41,44)(45,47)(48,51)(53,57)(54,56)(58,61)(62,64)

Polycyclic presentation:

2^10|a^2=d,b^2=ef,d^2=j,b^a=be,c^a=cg,c^b=cj,d^b=di,d^c=dh,e^a=ei,e^b=ei,e^c=ehj,f^b=fi,f^c=fhj,g^a=gh,g^b=gh,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342829, #72342849, #72342855
maximal normal subgroup of	#72342829, #72342849, #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.5*F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
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);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);