#72319602

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	#72317797
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318243	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

4133 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)(5,24)(7,27)(9,19)(10,36)(15,47)(17,50)(21,25)(22,23)(28,60)(29,42)(32,53)(33,52)(34,35)(37,56)(38,39)(41,43)(51,61)(57,58)(62,63);(1,2,6,12)(3,13,16,8)(4,19,22,35)(5,10,24,36)(7,11)(9,18,34,23)(14,43,44,41)(15,42,47,29)(17,30)(20,53,54,32)(21,56,57,33)(25,52,58,37)(26,39)(27,40)(28,46,60,45)(31,50)(38,59)(48,51,55,61)(49,63)(62,64);(1,3)(2,8)(4,20,18,48)(5,15)(6,16)(7,17)(9,32,35,61)(10,29)(11,31)(12,13)(14,21,46,25)(19,51,34,53)(22,54,23,55)(24,47)(26,49)(27,50)(28,33,41,37)(30,40)(36,42)(38,63)(39,62)(43,52,60,56)(44,57,45,58)(59,64);(1,4)(2,9)(3,14)(5,25)(6,18)(7,23)(8,28)(10,37)(11,19)(12,35)(13,41)(15,48)(16,46)(17,45)(20,47)(21,24)(22,27)(26,57)(29,61)(30,60)(31,43)(32,42)(33,36)(34,40)(38,52)(39,56)(44,50)(49,54)(51,63)(53,62)(55,64)(58,59)

Polycyclic presentation:

2^11|b^2=gh,c^2=hk,e^2=k,i^2=k,b^a=bg,c^a=chk,c^b=ck,d^a=df,d^b=dfhjk,d^c=de,e^a=eh,e^b=ehk,e^d=ek,f^b=fhik,f^c=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

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320243
maximal normal subgroup of	#72320243

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*F.8);;
gap> SetPower(coll, 3, F.8*F.11);;
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.8*F.10*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, 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);