#72286660

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#12966
commutator	#124	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	#60639	C₂²
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,d^2=ij,e^2=ij,h^2=j,b^a=bg,c^a=ce,c^b=cei,d^a=dj,d^b=dhij,d^c=dij,e^a=ei,e^b=eh,e^c=eij,f^c=fj,g^c=ghij,g^d=gij,g^e=gi,h^a=hi,h^b=hij,h^c=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286685, #72286867, #72287068
maximal normal subgroup of	#72286685, #72286867, #72287068

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