#72225548

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#96692
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×#346	C₂³
socle	C₂	#72216546
prime radical	C₂×#135	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#346 ◁ C₄×#1313 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

867 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=d,d^2=g,e^2=h,f^2=ik,h^2=j,i^2=k,j^2=k,b^a=bf,c^a=cik,c^b=chj,d^a=dk,d^b=dj,e^c=ej,e^d=ek,f^a=fi,f^b=fi,f^c=fk,g^b=gk,h^c=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225554, #72227167, #72227171
maximal normal subgroup of	#72225554, #72227167, #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.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, 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);