#72227090

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216588
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#490484	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=f,c^2=djk,d^2=g,e^2=h,f^2=i,h^2=j,i^2=k,j^2=k,b^a=be,c^a=chjk,c^b=cef,d^b=djk,d^c=dk,e^a=eh,e^b=ehj,e^c=ehk,e^d=ek,f^a=fk,f^c=fi,g^b=gk,h^a=hj,h^b=hjk,h^c=hj,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227091, #72227167, #72227175
maximal normal subgroup of	#72227091, #72227167, #72227175

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