#72226854

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226853
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	#72216275	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,c^2=fh,d^2=gi,e^2=ghij,f^2=h,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=be,c^a=cf,c^b=cfj,d^b=dg,d^c=dk,e^a=eh,e^b=egh,f^a=fhjk,f^b=fhj,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226858, #72226924, #72226936
maximal normal subgroup of	#72226858, #72226924, #72226936

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