#72313619

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	C₂×#509	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ #20521 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2501 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=f,d^2=f,e^2=h,h^2=i,i^2=j,b^a=bgj,c^a=cg,c^b=cgj,d^a=dg,d^b=dj,d^c=dg,e^c=eh,e^d=ei,f^e=fj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313732, #72314014, #72314619
maximal normal subgroup of	#72313732, #72314014, #72314619

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