#72314036

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	#7623475	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2521 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)(3,5,7,8)(9,10,13,16)(11,19,20,15)(12,17,24,29)(14,18,25,30)(21,34,35,27)(22,32,36,28)(23,33,26,31)(37,39,38,40);(1,3,4,7)(2,5,6,8)(9,11,13,20)(10,15,16,19)(12,21,24,35)(14,22,25,36)(17,27,29,34)(18,28,30,32)(23,38,26,37)(31,40,33,39);(2,6)(5,8)(10,16)(11,20)(12,23,24,26)(14,22)(17,31,29,33)(18,28)(21,37,35,38)(25,36)(27,39,34,40)(30,32);(2,6)(3,7)(10,16)(12,21,24,35)(14,25)(15,19)(17,27,29,34)(22,36)(23,37,26,38)(31,39,33,40);(9,12,22,37,20,35,14,26,13,24,36,38,11,21,25,23)(10,17,32,39,15,27,18,31,16,29,28,40,19,34,30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314270, #72315244
maximal normal subgroup of	#72314098, #72314270, #72315244

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