#72258134

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#12067
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
socle	C₂	C₂×#122963
prime radical	#33981	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ #149502 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

797 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:

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

Polycyclic presentation:

2^11|a^2=k,c^2=e,d^2=i,e^2=g,f^2=hjk,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=cf,c^b=cfh,d^a=djk,d^b=djk,e^a=eh,e^b=ehjk,f^a=fh,f^b=fh,f^c=fj,f^e=fk,g^a=gj,g^b=gjk,h^a=hjk,h^b=hjk,h^c=hk,i^a=ik,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258148
maximal normal subgroup of	#72258148

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