#72227183

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#11999
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂³
socle	C₂	#72227182
prime radical	C₂³	#12002
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

475 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=eik,c^2=f,d^2=g,e^2=h,f^2=ik,g^2=i,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cijk,d^b=djk,d^c=dk,e^a=eh,e^d=ek,g^b=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227186, #72227204, #72227228, …
maximal normal subgroup of	#72227186, #72227204, #72227228, …

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