#72287053

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#146566
commutator	C₂×#41	C₂×C₄²
Frattini, cosocle	C₂×#532	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2887 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)(3,14,17,33,19,9,18,32)(4,20,22,50)(5,24)(6,12)(7,28)(8,13)(10,21,34,53)(11,25)(15,41,43,62,44,49,48,63)(16,40,45,59,46,31,47,58)(23,51,29,52)(26,37)(27,36)(30,56,57,42,39,61,60,64)(35,54,38,55);(1,3,6,17)(2,9,13,14)(4,15,23,43)(5,16,26,47)(7,18,8,19)(10,30,34,39)(11,31)(12,33)(20,49)(21,42,54,56)(22,44,29,48)(24,40)(25,46,27,45)(28,32)(35,57)(36,58,37,59)(38,60)(41,50)(51,62,52,63)(53,64,55,61);(1,4)(2,10)(3,15)(5,21)(6,23)(7,29)(8,22)(9,30)(11,20)(12,38)(13,34)(14,39)(16,42)(17,43)(18,48)(19,44)(24,50)(25,53)(26,54)(27,55)(28,35)(31,49)(32,57)(33,60)(36,52)(37,51)(40,41)(45,61)(46,64)(47,56)(58,63)(59,62)

Polycyclic presentation:

2^10|a^2=e,b^2=g,e^2=ij,h^2=j,b^a=bef,c^a=cd,d^a=di,d^b=di,e^b=eh,e^c=ei,f^a=fhj,f^b=fi,f^c=fj,f^d=fj,f^e=fij,g^a=ghi,g^d=gj,g^e=gij,g^f=gj,h^a=hi,h^b=hi,h^c=hj,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287068, #72287073, #72287086, …
maximal normal subgroup of	#72287068, #72287073, #72287086, …

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