#72232005

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#2177
commutator	C₁₆	C₂⁴×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁵
socle	C₂²	C₂²×#2177
prime radical	C₂×#33981	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2,5,4)(3,6,8,7)(9,10)(11,21)(12,28)(13,33)(14,36)(15,16)(17,25)(18,31)(19,30)(20,34)(22,45)(23,38)(24,46)(26,59)(27,60)(29,40)(32,65)(35,66)(37,49)(39,63)(41,56)(42,70)(43,54)(44,52)(47,58)(48,57)(50,71)(51,69)(53,61)(55,68)(62,64)(67,72);(11,22)(17,37)(21,45)(24,27)(25,49)(26,50)(41,43)(42,67)(46,60)(47,69)(48,68)(51,58)(54,56)(55,57)(59,71)(70,72);(10,16)(17,37)(18,38)(19,39)(20,36)(21,45)(28,40)(33,66)(41,43)(42,67)(44,61)(46,60)(47,69)(48,68)(59,71)(62,65);(9,11,23,51,13,26,32,56,34,24,53,70,63,55,12,25,15,22,31,58,35,50,64,54,14,27,52,72,30,57,29,49)(10,17,40,68,19,42,44,46,36,41,62,59,66,69,18,21,16,37,28,48,39,67,61,60,20,43,65,71,33,47,38,45);(1,3,4,7,5,8,2,6)(11,24,22,27)(12,29)(17,41,37,43)(18,38)(21,46,45,60)(23,31)(25,56,49,54)(26,55,50,57)(28,40)(32,64)(42,69,67,47)(44,61)(48,59,68,71)(51,72,58,70)(52,53)(62,65)

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=ck,d^a=df,d^b=dk,e^d=ej,f^a=fh,f^e=fk,g^d=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

272 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.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, 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);