#72319364

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72319010
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	#2108	C₂⁴
prime radical	#72287059	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622661 ◁ G
LECS	G ▷ #2108 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5024 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:

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

Polycyclic presentation:

2^11|b^2=e,d^2=j,e^2=h,g^2=ijk,h^2=jk,i^2=j,j^2=k,b^a=bg,c^b=cf,d^a=djk,d^b=dk,d^c=dhik,e^a=ei,e^c=ek,f^a=fhik,f^b=fk,f^d=fk,g^a=gi,g^b=gj,g^c=ghi,g^e=gk,g^f=gk,h^a=hj,i^a=ijk,i^b=ik,j^a=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319398
maximal normal subgroup of	#72319398

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