#72286601

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#147826
commutator	#532	C₂²×C₄
Frattini, cosocle	C₂×#532	C₂³
prime radical	C₂×#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1798 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)(3,8)(4,18,21,48)(5,24)(6,28)(7,11)(9,20,35,52)(10,26)(12,29)(13,41,43,61)(14,31)(15,40)(16,32)(17,33)(19,50,44,64)(22,49,23,55)(25,38)(27,39)(30,42,59,45)(34,63,60,51)(36,53,37,54)(46,56)(47,62)(57,58);(1,3)(2,8)(4,19,22,43)(5,25,26,46)(6,15,29,16)(7,17)(9,34,36,30)(10,38,24,56)(11,33)(12,40,28,32)(13,21,44,23)(14,27)(18,41,49,64)(20,45,53,51)(31,39)(35,60,37,59)(42,54,63,52)(47,58)(48,61,55,50)(57,62);(1,4,7,21)(2,9,11,35)(3,13)(5,20,26,52)(6,22,29,23)(8,30)(10,18,24,48)(12,36,28,37)(14,45)(15,19)(16,44)(17,43)(25,51)(27,54,58,53)(31,61)(32,34)(33,59)(38,50)(39,55,57,49)(40,60)(41,62)(42,47)(46,63)(56,64)

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=i,d^2=ij,e^2=ij,h^2=j,b^a=bfg,c^a=cd,c^b=ce,d^a=di,d^b=dh,d^c=dij,e^a=ej,e^b=ehi,e^c=ei,f^c=fj,g^c=gh,g^d=gi,g^e=gij,h^a=hi,h^b=hij,h^c=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286647, #72286910, #72287007
maximal normal subgroup of	#72286647, #72286910, #72287007

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