#72321847

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 ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2350 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,e^2=hj,f^2=h,g^2=i,i^2=j,b^a=bgij,d^a=dhj,e^a=eghj,e^c=ej,e^d=ehj,f^a=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321949, #72343989
maximal normal subgroup of	#72321949, #72343989

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