#72260437

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#2177
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁵
prime radical	#72260425	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479943 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^11|a^2=i,d^2=fhk,e^2=g,f^2=hj,g^2=i,h^2=jk,j^2=k,c^a=ci,c^b=ci,d^a=df,d^b=dk,d^c=dk,e^d=ej,f^a=fh,f^e=fk,g^d=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 4, F.6*F.8*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);