#72313369

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	D₄×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20493 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3316 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:

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

Polycyclic presentation:

2^10|b^2=gj,c^2=f,d^2=f,e^2=gh,h^2=i,i^2=j,b^a=bg,c^b=cg,d^a=dg,d^b=dj,e^a=eg,e^c=eh,e^d=egi,f^e=fj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313561, #72313898, #72314098, …
maximal normal subgroup of	#72313561, #72313898, #72314098, …
maximal quotient of	#72343673, #72343678

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