#72225326

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#11999
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225325
prime radical	C₂³	#12668
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=dhj,c^2=f,d^2=g,e^2=hk,f^2=i,g^2=k,h^2=j,j^2=k,b^a=be,c^a=chi,c^b=chi,d^b=dj,e^a=eh,e^b=eh,e^d=ek,f^a=fj,f^b=fj,g^b=gk,h^a=hj,h^b=hj,i^a=ik,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227105, #72227121, …
maximal normal subgroup of	#72225336, #72227105, #72227121, …

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