#72315166

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#532	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2823 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,28)(20,25)(21,29)(22,30)(23,26)(24,27)(31,36)(32,39)(33,37)(34,40)(35,38);(2,5)(3,7)(10,15)(16,25)(17,26)(18,27)(28,36)(29,37)(30,38)(39,40);(1,3)(2,6)(4,7)(5,8)(10,15)(16,25)(17,26)(18,27)(28,36)(29,37)(30,38)(39,40);(3,7)(6,8)(11,19,20,31)(12,22)(14,24)(16,28,25,36)(17,30)(18,27)(21,34,33,32)(23,35)(26,38)(29,40,37,39);(9,11,12,19,14,21,22,32,13,20,23,31,24,33,35,34)(10,16,17,28,18,29,30,39,15,25,26,36,27,37,38,40)

Polycyclic presentation:

2^10|d^2=f,e^2=g,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=chj,d^c=dhj,e^d=eg,f^e=fj,g^d=gi,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315187, #72315361, #72315522
maximal normal subgroup of	#72315187, #72315361, #72315522

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