#72284293

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235677
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72283864	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35616 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4416 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)(5,23)(8,27)(9,18)(14,42)(15,44)(16,40)(17,41)(19,39)(24,53)(28,57)(29,32)(31,36)(35,38)(43,61)(45,63)(46,56)(47,58)(49,64)(50,62);(1,2)(3,14)(4,18)(5,10)(6,11,26,37)(7,12)(8,16)(9,22)(13,38,41,42)(15,43,19,49)(17,35)(20,30)(21,36,25,31)(23,34)(24,54,53,33)(27,56,32,40)(28,45,47,50)(29,46)(39,61)(44,64)(48,52)(51,60,55,59)(57,62)(58,63);(1,3)(2,8,12,29)(4,19)(5,13,23,41)(6,16,26,46)(7,17)(9,28,18,47)(10,32)(11,35)(14,37)(15,22)(20,44,52,39)(21,45,25,50)(24,40)(27,34)(30,58)(31,43)(33,42,54,38)(36,49)(48,57)(51,62)(53,56)(55,63)(59,64,60,61);(1,4,7,22)(2,9)(3,15)(5,20)(6,25)(8,28,29,47)(10,30,34,48)(11,36,37,31)(12,18)(13,39,41,44)(14,43)(16,45,46,50)(17,19)(21,26)(23,52)(24,51,53,55)(27,57)(32,58)(33,59)(35,49)(38,61,42,64)(40,63)(54,60)(56,62)

Polycyclic presentation:

2^11|b^2=gik,c^2=hi,d^2=hijk,e^2=h,f^2=hk,b^a=bg,c^a=ce,c^b=cf,d^a=dghk,d^b=dgj,d^c=dhk,e^a=eh,e^b=ei,e^c=ehj,e^d=eijk,f^b=fhi,f^c=fhj,f^d=fij,g^b=gj,g^c=gik,g^d=gjk,g^e=gjk,g^f=gj,h^b=hk,h^d=hk,i^a=ij,i^b=ij,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285855
maximal normal subgroup of	#72285855

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