#72343103

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	#72318976
commutator	C₂×M₃₂	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	#3444	D₈
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

1545 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=efg,d^2=h,e^2=ik,g^2=ijk,i^2=j,j^2=k,c^a=ce,c^b=cf,d^c=dj,e^a=ei,e^b=eh,e^c=eijk,e^d=ek,f^a=fhk,f^c=fh,f^e=fk,g^a=ghijk,g^c=ghi,g^d=gk,g^e=gk,h^c=hk,i^a=ij,i^c=ijk,j^a=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343118, #72343317, #72343385
maximal normal subgroup of	#72343118, #72343317, #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.4);;
gap> SetPower(coll, 3, F.5*F.6*F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9*F.11);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
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, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, 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, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);