#72313761

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	#7623358	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20473 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2915 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fgi,d^2=fghj,e^2=g,h^2=i,i^2=j,b^a=bj,c^a=cg,c^b=cghij,d^a=dgj,d^b=dghi,d^c=dhj,e^d=eg,f^c=fj,f^d=fj,h^b=hi,h^c=hi,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313816, #72313898, #72315286
maximal normal subgroup of	#72313816, #72313898, #72315286

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