#72287644

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72287583
commutator	C₂×M₃₂	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	#3444	SD₁₆
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622660 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=h,b^2=d,c^2=efgi,d^2=h,e^2=k,f^2=ij,i^2=j,j^2=k,b^a=bh,c^a=ce,c^b=cefik,d^c=djk,e^b=ehk,f^a=fhk,f^b=fhijk,f^c=fk,f^d=fk,f^e=fk,g^a=gh,g^b=gk,g^c=gk,g^e=gk,g^f=gk,h^c=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287845, #72343317, #72343423
maximal normal subgroup of	#72287845, #72343317, #72343423

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