#72321998

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	#33526	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2327 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,27,20,28,33,43,22,26,36,25,34,41,48,42)(19,24,31,23,32,38,35,39,29,40,45,37,30,46,47,44);(1,3,4,10,5,9,11,15)(2,6,8,12,7,13,14,16)(17,19)(18,23,26,37)(20,30)(21,35)(22,29)(24,27,40,25)(28,44,41,39)(31,33)(32,34)(36,47)(38,42,46,43)(45,48);(17,19)(18,24)(20,32)(21,31)(22,29)(23,27)(25,37)(26,40)(28,38)(30,34)(33,35)(36,45)(39,43)(41,46)(42,44)(47,48);(2,7)(3,9)(8,14)(10,15)(19,29)(23,37)(24,40)(30,32)(31,45)(35,47)(38,46)(39,44);(2,7)(3,9)(8,14)(10,15)(17,20)(18,25)(19,30)(22,34)(23,24)(26,27)(28,43)(29,32)(31,45)(33,48)(37,40)(38,44)(39,46)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322136
maximal normal subgroup of	#72322136
maximal quotient of	#72343655, #72343670

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, 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);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*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);