#72318311

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
prime radical	#72241309	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

4755 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:

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

Polycyclic presentation:

2^11|b^2=hij,c^2=gk,d^2=hk,i^2=k,b^a=be,c^a=cg,c^b=cf,d^b=defhij,d^c=dk,e^b=ej,e^c=ei,e^d=eh,f^a=fhjk,f^b=fjk,f^c=fhij,f^d=fh,g^b=ghijk,g^e=gjk,g^f=gj,h^b=hk,i^a=ij,i^c=ij,i^e=ik,i^f=ik,i^g=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

68 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.8*F.9*F.10);;
gap> SetPower(coll, 3, F.7*F.11);;
gap> SetPower(coll, 4, F.8*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, 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, 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, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);