#72313637

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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=f,d^2=f,e^2=fghi,h^2=i,i^2=j,b^a=bj,c^b=cg,d^a=dg,d^b=dgj,e^b=eg,e^c=eh,e^d=egi,f^e=fj,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313732, #72314098, #72314574
maximal normal subgroup of	#72313732, #72314098, #72314574

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