#72318043

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#128033
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂×#536	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2247 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dgh,c^2=efj,e^2=j,f^2=j,g^2=j,h^2=j,b^a=bgh,c^a=ce,c^b=ceg,d^c=dg,e^a=eg,e^b=egi,e^c=eghj,e^d=eij,f^a=fhj,f^b=fhi,f^c=fghj,f^d=fij,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,g^e=gj,g^f=gj,h^a=hi,h^b=hij,h^d=hj,h^e=hj,h^f=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318072, #72318175, #72318208
maximal normal subgroup of	#72318072, #72318175, #72318208

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