#72321983

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	#7623254	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2229 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,d^a=dj,d^b=dj,d^c=dj,e^a=eg,e^c=ej,e^d=eh,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, #72322814, #72322909
maximal normal subgroup of	#72322136, #72322814, #72322909
maximal quotient of	#72343641, #72343674

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