#72284595

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72234509
commutator, Frattini, cosocle	#2308	C₂⁴
LCS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ #35448 ◁ G
LECS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

5927 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)(3,13,17,35)(4,18)(5,10)(6,11)(7,12)(8,16,31,22)(9,21)(14,42,45,40)(15,41,48,58)(19,33)(20,52)(23,51)(24,36)(25,37)(26,38)(27,34)(28,39)(29,44,56,49)(30,47,32,50)(43,62,46,61)(53,59)(54,60)(55,64,57,63);(3,14,17,45)(6,26)(8,29,31,56)(11,38)(13,40,35,42)(15,46,48,43)(16,49,22,44)(24,28)(30,55,32,57)(36,39)(41,62,58,61)(47,63,50,64);(1,3,7,17)(2,8,12,31)(4,15,21,48)(5,22,25,16)(6,14,26,45)(9,32,18,30)(10,35,37,13)(11,29,38,56)(19,47,23,50)(20,43,27,46)(24,44,28,49)(33,58,51,41)(34,55,52,57)(36,42,39,40)(53,64,54,63)(59,62,60,61);(1,4)(2,9,12,18)(3,15)(5,23,25,19)(6,27,26,20)(7,21)(8,30,31,32)(10,33)(11,34)(13,41)(14,43,45,46)(16,50,22,47)(17,48)(24,54)(28,53)(29,55)(35,58)(36,60,39,59)(37,51)(38,52)(40,61,42,62)(44,63)(49,64)(56,57)

Polycyclic presentation:

2^11|a^2=jk,b^2=jk,c^2=k,d^2=hik,e^2=hk,f^2=h,b^a=bjk,c^a=ce,c^b=cf,d^a=dg,d^b=dgh,d^c=dhik,e^a=eh,e^b=ejk,e^c=ehk,e^d=ei,f^a=fjk,f^b=fhk,f^c=fh,f^d=fi,f^e=fk,g^c=gik,g^d=gj,g^e=gjk,g^f=gj,h^d=hk,i^a=ij,i^b=ijk,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	#72286327
maximal normal subgroup of	#72286327

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