#72340474

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#151196
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72236508	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

4382 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)(3,13)(4,18,22,53)(5,23,25,11)(6,26)(7,9)(8,16,32,44)(10,34)(12,29)(14,37)(15,45,19,55)(17,49)(20,54)(21,35,59,38)(24,41,28,39)(27,43)(30,50,61,47)(31,48,40,51)(33,57)(36,58)(42,63)(46,52)(56,64)(60,62);(1,3)(2,8,9,32)(4,19,22,15)(5,16,25,44)(6,27,29,17)(7,14)(10,30)(11,37,23,13)(12,40)(18,52,53,64)(20,56)(21,47)(24,48)(26,31)(28,51)(33,61)(34,62,57,63)(35,42)(36,55,54,45)(38,60)(39,43)(41,49)(46,58)(50,59);(2,9)(3,14)(10,33)(11,23)(12,26)(15,19)(16,44)(17,27)(18,53)(35,38)(36,54)(39,41)(46,56)(47,50)(48,51)(62,63);(1,4)(2,10,9,33)(3,15,14,19)(5,20,25,58)(6,21,29,59)(7,22)(8,30,32,61)(11,38)(12,36)(13,42)(16,46)(17,50)(18,39,53,41)(23,35)(24,57)(26,54)(27,47)(28,34)(31,55)(37,60)(40,45)(43,52,49,64)(44,56)(48,63,51,62)

Polycyclic presentation:

2^11|a^2=i,b^2=gj,d^2=hi,e^2=hj,f^2=k,b^a=be,c^a=ck,c^b=ck,d^a=defk,d^b=dhjk,e^a=eh,e^b=ehij,e^d=eg,f^a=fik,f^b=fghj,f^d=fghijk,f^e=fi,g^a=gi,g^d=gk,g^e=gjk,g^f=gk,h^a=hk,h^b=hk,h^d=hjk,h^f=hk,i^b=ij,i^d=ijk,i^e=ik,j^a=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340478
maximal normal subgroup of	#72340478

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