#72287447

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	#7623529	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2202 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=ghij,d^2=f,e^2=fhij,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=chj,c^b=ch,d^a=dhj,d^b=di,d^c=dhi,e^a=ej,e^b=egi,e^c=egh,e^d=eij,f^b=fj,f^c=fj,f^e=fj,g^d=gj,g^e=gi,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287517, #72319103
maximal normal subgroup of	#72287517, #72319103

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