#72287289

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	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,26)(20,36)(21,29)(22,30)(23,31)(24,32)(25,28)(27,39)(33,42)(34,43)(35,44)(37,46)(38,45)(40,48)(41,47);(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19,21,20,23,33,41,37,25,35,22,39,24,34,40,38)(18,26,29,36,31,42,47,46,28,44,30,27,32,43,48,45);(2,7)(3,10)(8,14)(11,16)(17,20,22,33,23,37,40,35,25,39,21,34,24,38,41,19)(18,27,30,43,31,45,48,26,28,36,29,42,32,46,47,44);(2,7)(6,9)(8,14)(13,15)(18,28)(19,20)(21,40)(22,41)(23,24)(26,27)(29,47)(30,48)(33,38)(34,37)(35,39)(36,44)(42,46)(43,45)

Polycyclic presentation:

2^10|c^2=f,d^2=g,f^2=h,g^2=ij,h^2=j,i^2=j,c^b=chi,d^a=dj,d^b=dhi,d^c=dh,e^a=ej,e^b=ehi,e^c=egi,e^d=eg,f^d=fj,f^e=fi,g^c=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287390, #72287479, #72320987
maximal normal subgroup of	#72287390, #72287479, #72320987

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