#72284923

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	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
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

3610 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:

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

Polycyclic presentation:

2^11|b^2=ghk,c^2=i,d^2=hik,e^2=h,f^2=k,b^a=bghk,c^a=ce,c^b=cf,d^a=dgk,d^b=dgjk,d^c=dhijk,e^a=eh,e^b=eg,e^c=ehj,e^d=eik,f^a=fgk,f^b=fik,f^d=fij,f^e=fij,g^b=gjk,g^c=gi,g^d=gj,g^e=gjk,h^b=hjk,h^d=hk,h^f=hk,i^a=ij,i^b=ik,i^d=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

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