#72319836

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#182043
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₄×M₃₂	C₂⁴
socle	C₂	#72285281
prime radical	#3444	M₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ #479908 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1764 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:

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

Polycyclic presentation:

2^11|a^2=g,b^2=e,d^2=j,e^2=h,g^2=i,i^2=j,j^2=k,b^a=bijk,c^a=cf,d^b=dk,d^c=dh,e^a=ej,f^a=fhk,f^d=fk,g^b=gj,g^c=ghk,g^e=gk,g^f=gk,h^a=hk,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319914, #72343292
maximal normal subgroup of	#72319914, #72343292

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