#72225509

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#495812
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	#72225429	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^11|c^2=f,d^2=g,e^2=hj,g^2=i,h^2=jk,j^2=k,b^a=be,c^a=ck,c^b=cg,d^a=dj,d^b=dgi,e^a=eh,e^b=eh,e^c=ek,f^b=fi,g^a=gk,g^b=gi,h^a=hj,h^b=hj,j^a=jk,j^b=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225549
maximal normal subgroup of	#72225549
maximal quotient of	#72225510

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