#72340388

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#151342
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72285472	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁶ ◁ C₂×#1781 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3928 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)(3,8)(4,17)(5,11)(6,12)(7,9)(10,21)(13,30)(14,42)(15,45)(16,29)(18,49)(19,37)(20,38)(22,56)(23,31)(24,35)(25,32)(26,33)(27,44)(28,40)(34,50)(36,53)(39,63)(41,60)(43,58)(46,59)(47,51)(48,52)(54,61)(55,62)(57,64);(1,3)(2,8)(4,18)(5,15)(6,16,26,41)(7,13)(9,30)(10,34)(11,28)(12,29,33,60)(14,22,39,52)(17,47,37,49)(19,51)(20,43,24,46)(21,54,23,50)(25,40)(27,36,57,62)(31,61)(32,45)(35,58,38,59)(42,56)(44,53)(48,63)(55,64);(2,9)(3,13)(4,19)(11,32)(12,33)(15,40)(16,41)(21,23)(22,52)(27,57)(34,61)(35,38)(42,63)(43,46)(47,49)(53,55);(1,4,20,53)(2,10,35,56)(3,14,43,54)(5,23,26,52,25,21,6,22)(7,19,24,55)(8,27,58,47)(9,31,38,48)(11,37,33,62,32,17,12,36)(13,39,46,50)(15,44,41,51,40,64,16,18)(28,42,60,61,45,63,29,34)(30,57,59,49)

Polycyclic presentation:

2^11|b^2=ghij,d^2=efgik,e^2=h,f^2=k,b^a=bgjk,c^a=ck,c^b=ck,d^a=de,d^b=df,d^c=dk,e^a=eh,e^b=eg,e^d=egk,f^a=fgk,f^b=fi,f^d=fgij,f^e=fij,g^b=gj,g^d=gi,g^e=gjk,h^b=hjk,h^d=hjk,h^f=hk,i^a=ij,i^b=ik,i^d=ijk,i^e=ik,j^d=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340399
maximal normal subgroup of	#72340399

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