#72286252

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285353
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#7623252	C₂²
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #479960 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1585 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=h,c^2=ehik,e^2=h,f^2=ik,g^2=ij,i^2=j,j^2=k,c^a=cf,c^b=cg,d^a=dk,d^b=dh,d^c=dk,e^c=ej,f^a=fi,f^b=fhij,f^c=fik,f^e=fk,g^a=ghijk,g^b=gi,g^c=gij,g^d=gk,g^e=gk,g^f=gk,h^c=hk,i^a=ij,i^b=ijk,i^c=ij,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287481, #72343385
maximal normal subgroup of	#72287481, #72343385

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