#72267557

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	#890702	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

16655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(26,30)(31,38)(32,36)(39,40);(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,23,24,20)(27,33)(28,35)(31,38)(32,36);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(17,20,21,23)(18,19,22,24)(25,26,29,30)(27,31,33,38)(28,36,35,32)(34,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27,29,33)(26,31,30,38)(28,37,35,34)(32,40,36,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,35)(26,32,30,36)(27,34,33,37)(31,39,38,40)

Polycyclic presentation:

2^10|c^2=ghi,d^2=ij,e^2=j,f^2=j,c^a=cg,c^b=ch,d^a=dgh,d^b=dgj,d^c=dhi,e^c=ej,e^d=ei,f^c=fij,f^d=fj,f^e=fj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267566

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