#72285482

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	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2510 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,d^2=fi,e^2=h,f^2=h,g^2=i,i^2=j,b^a=bg,c^a=cj,d^a=di,d^b=dhij,e^b=eh,f^b=fj,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285671, #72287099, #72287479, …
maximal normal subgroup of	#72285671, #72287099, #72287479, …

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