#72314534

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	#7623471	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2620 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=hj,e^2=fghj,g^2=ij,i^2=j,c^a=ch,c^b=chj,d^b=dhj,e^a=ehij,e^b=ej,e^c=eg,e^d=egj,f^e=fj,g^a=gj,g^c=gi,g^d=gi,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314559, #72314594, #72315244
maximal normal subgroup of	#72314559, #72314594, #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.6);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);