#72227235

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#7175
commutator	C₂×C₁₆	C₈²
Frattini, cosocle	C₈×#346	C₂²
socle	C₂	#72227234
prime radical	C₂³	#7162
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=e,c^2=f,d^2=gj,e^2=h,f^2=j,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bd,c^b=cij,d^a=dg,d^b=dgi,d^c=dk,e^a=ejk,e^c=ek,g^a=gi,g^b=gik,h^a=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227239, #72227250, #72227267, …
maximal normal subgroup of	#72227239, #72227250, #72227267, …

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