#72225547

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	#72225546
prime radical	C₂×#135	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#346 ◁ C₄×#1398 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 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:

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

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=cf,c^b=cfhij,d^a=di,d^b=dijk,e^c=ej,e^d=ek,f^a=fi,f^b=fi,f^c=fk,g^a=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, #72227166, #72227168
maximal normal subgroup of	#72225554, #72227166, #72227168

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