#72225330

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#128688
commutator	C₂×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72216549
prime radical	C₂×#508	C₄²
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

845 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=f,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=chj,c^b=cijk,d^a=djk,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

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225336, #72227166, #72227171
maximal normal subgroup of	#72225336, #72227166, #72227171

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