#72225333

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	C₄	#129165
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72225332
prime radical	C₂³	#12678
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#1313 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=fk,c^2=d,d^2=g,e^2=hj,f^2=i,h^2=jk,i^2=k,j^2=k,b^a=be,c^a=chij,c^b=cij,d^a=dj,e^a=eh,e^b=eh,e^c=ej,e^d=ek,f^c=fk,g^a=gk,h^a=hj,h^b=hj,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227175, #72227179
maximal normal subgroup of	#72225336, #72227175, #72227179

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