#72340929

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	#72237744
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72284399	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20585 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

2149 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)(3,15)(4,21,24,59)(5,25)(6,30,27,12)(7,13,35,47)(8,14)(9,18,39,56)(10,23)(11,29)(16,37,22,43)(17,53)(19,57,32,50)(20,51)(26,49)(28,41,60,45)(31,44)(33,42)(34,40)(36,61,64,54)(38,62)(46,55)(48,52)(58,63);(1,3)(2,9,14,39)(4,22)(5,26,29,17)(6,18,27,56)(7,32,35,19)(8,20)(10,36,42,64)(11,43)(12,15)(13,46,47,38)(16,24)(21,53,59,49)(23,58,33,52)(25,37)(28,61)(30,51)(31,62)(34,55)(40,50)(41,48)(44,57)(45,63)(54,60);(1,4)(2,10)(3,16)(5,27,29,6)(7,33,35,23)(8,24)(9,36)(11,44,25,40)(12,45,30,41)(13,21)(14,42)(15,48,51,63)(17,18,26,56)(19,58,32,52)(20,22)(28,31)(34,60)(37,57,43,50)(38,49)(39,64)(46,53)(47,59)(54,62)(55,61);(1,5)(2,11)(3,17)(4,6)(7,28)(8,29)(9,37)(10,40)(12,21)(13,45)(14,25)(15,49)(16,18)(19,54)(20,26)(22,56)(23,31)(24,27)(30,59)(32,61)(33,34)(35,60)(36,50)(38,48)(39,43)(41,47)(42,44)(46,63)(51,53)(52,55)(57,64)(58,62)

Polycyclic presentation:

2^11|a^2=ghij,b^2=hj,c^2=h,f^2=ghjk,b^a=be,c^a=cf,c^b=chjk,d^a=dk,d^b=dik,d^c=dghjk,e^a=ej,e^b=ei,e^c=eg,f^a=fgh,f^b=fgj,f^c=fghijk,f^e=fijk,g^a=gi,g^b=gjk,g^c=gk,g^d=gk,g^f=gjk,h^a=hik,h^d=hk,h^e=hk,h^f=hjk,i^b=ik,i^c=ij,i^f=ik,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341003
maximal normal subgroup of	#72341003

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