#72318432

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72233443
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72216832	C₂
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35327 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4601 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:

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

Polycyclic presentation:

2^11|b^2=gk,c^2=hk,d^2=efik,i^2=k,b^a=bg,c^b=ck,d^a=de,d^b=df,d^c=defij,e^b=ei,e^c=eh,e^d=ehk,f^a=fhjk,f^b=fhij,f^c=fh,f^d=fh,g^d=ghijk,g^e=gjk,g^f=gj,h^d=hk,i^a=ij,i^b=ij,i^d=ik,i^e=ik,i^f=ik,i^g=ik,j^b=jk,j^d=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319061
maximal normal subgroup of	#72319061

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