#72343253

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	#7623106	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2879 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=g,f^2=j,h^2=i,i^2=j,b^a=bi,c^a=cg,d^c=dj,e^a=eh,e^b=eh,e^c=eg,e^d=eg,f^e=fj,h^a=hi,h^b=hij,h^e=hij,i^a=ij,i^b=ij,i^e=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343283
maximal normal subgroup of	#72343283
maximal quotient of	#72343766

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