#72286949

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	#7624012	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2282 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)(9,12)(10,13)(11,14)(15,16)(17,18,21,24,22,28,35,43,23,26,38,42,39,45,33,46)(19,29,20,32,31,47,36,44,30,48,34,27,40,41,37,25);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(18,24,26,42)(19,20,30,34)(21,35)(22,39)(27,32)(28,46,45,43)(29,47)(31,37,40,36)(33,38)(41,48);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19)(18,25)(20,33)(21,37)(22,40)(23,30)(24,41)(26,44)(27,28)(29,46)(31,39)(32,45)(34,35)(36,38)(42,47)(43,48);(2,7)(6,12)(8,14)(13,16)(18,26)(19,30)(20,34)(24,42)(28,45)(31,40)(36,37)(43,46);(3,9)(6,12)(10,15)(13,16)(17,20,35,30,39,37,21,31,23,34,33,19,22,36,38,40)(18,27,43,29,45,44,24,41,26,32,46,48,28,25,42,47)

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=gi,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=cg,c^b=cg,d^a=dhj,d^c=dhj,e^a=ej,e^b=eghi,e^c=egj,e^d=ej,f^a=fj,f^c=fj,f^e=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287030, #72287657, #72287809, …
maximal normal subgroup of	#72287030, #72287657, #72287809, …
maximal quotient of	#72343725, #72343776

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