#72287406

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285321
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#72287059	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622661 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5011 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:

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

Polycyclic presentation:

2^11|b^2=hj,c^2=e,d^2=ik,e^2=h,f^2=ijk,h^2=k,i^2=jk,j^2=k,c^a=cg,c^b=cf,d^a=dk,d^b=dijk,d^c=dik,e^b=ej,e^d=ek,f^a=fhjk,f^b=fi,f^c=fijk,f^e=fk,g^b=ghj,g^f=gk,h^b=hk,i^b=ij,i^c=ijk,j^b=jk,j^c=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287480
maximal normal subgroup of	#72287480

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