#72318267

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	#72317791
commutator, Frattini, cosocle	C₂×#535	C₂⁴
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

4825 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:

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

Polycyclic presentation:

2^11|b^2=gk,c^2=eghi,d^2=hk,f^2=j,i^2=k,b^a=bg,c^a=cf,c^b=ce,d^a=dhk,d^b=dh,d^c=defgik,e^a=eh,e^b=ehik,e^c=ehijk,e^d=eh,f^a=fj,f^b=fij,f^c=fh,f^d=fh,f^e=fjk,g^c=ghi,g^e=gjk,g^f=gj,h^c=hk,i^a=ij,i^b=ij,i^c=ij,i^e=ik,i^g=ik,j^b=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*F.11);;
gap> SetPower(coll, 3, F.5*F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.8*F.11);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
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.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, 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, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);