#72239792

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#499946	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,d^2=fhi,e^2=hj,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bh,d^a=df,d^b=dhj,d^c=dghj,e^a=ei,e^b=ej,f^a=fi,f^b=fj,f^c=fj,g^d=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239846
maximal normal subgroup of	#72239846

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