#72318226

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72317643
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318202	C₂
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35327 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=g,c^2=hk,d^2=ej,e^2=k,i^2=k,b^a=bg,c^a=chk,c^b=ck,d^a=df,d^b=dfk,d^c=de,e^a=eh,e^b=ehk,e^d=ek,f^b=fhik,f^c=fh,f^d=fh,f^e=fk,g^d=ghi,g^f=gjk,h^d=hk,i^a=ij,i^b=ij,i^f=ik,i^g=ik,j^b=jk,j^d=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319061
maximal normal subgroup of	#72319061

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