#72226889

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225208
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	#72225567	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2547 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)(3,15)(4,23)(5,14)(6,31)(7,10)(8,32)(9,12)(11,19)(13,24)(16,41)(17,49)(18,44)(20,42)(21,35)(22,43)(25,55)(26,37)(27,56)(28,54)(29,59)(30,40)(33,47)(34,58)(36,52)(38,48)(39,57)(45,63)(46,64)(50,60)(51,62)(53,61);(3,16)(4,24)(5,6)(7,8)(10,32)(11,35)(13,23)(14,31)(15,41)(17,45)(18,46)(19,21)(20,47)(22,48)(25,57)(26,27)(28,58)(29,30)(33,42)(34,54)(36,62)(37,56)(38,43)(39,55)(40,59)(44,64)(49,63)(50,53)(51,52)(60,61);(2,10,12,32)(3,17,18,22)(4,25)(11,36)(13,37,39,34)(14,31,40,59)(16,45,46,48)(19,47,52,50)(20,51,53,21)(23,54,55,56)(24,57)(26,58)(27,28)(33,60)(38,63)(41,64);(1,3,4,19,5,20,26,48,8,17,28,50,29,51,57,46,9,18,25,52,30,53,58,45,7,22,27,47,6,21,24,16)(2,11,37,49,59,64,55,61,10,33,13,15,14,38,54,62,12,36,34,43,31,41,23,42,32,60,39,44,40,63,56,35)

Polycyclic presentation:

2^11|c^2=e,d^2=fgh,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,c^a=cjk,c^b=cij,d^a=df,d^b=dfgi,d^c=dj,e^a=ek,e^d=ek,f^a=fh,f^b=fh,f^c=fk,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226910, #72227166, #72227167
maximal normal subgroup of	#72226910, #72227166, #72227167
maximal quotient of	#72227164

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