#72313868

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623358	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20476 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fgj,d^2=fgj,e^2=ghj,h^2=i,i^2=j,b^a=bj,c^a=cj,c^b=cj,d^a=dgj,d^b=dgj,d^c=dg,e^b=eh,e^c=egh,e^d=egi,f^e=fj,h^b=hi,h^c=hi,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313898, #72314251, #72314497
maximal normal subgroup of	#72313898, #72314251, #72314497

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