#72287124

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	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|c^2=g,d^2=f,f^2=h,g^2=i,i^2=j,c^b=chj,d^c=dghi,e^a=ej,e^b=ehj,e^c=eg,e^d=egj,f^c=fj,f^e=fj,g^d=gi,g^e=gij,i^d=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287479, #72319496
maximal normal subgroup of	#72287278, #72287479, #72319496

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