#72227072

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227052
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#13692	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1248 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:

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

Polycyclic presentation:

2^11|a^2=e,d^2=gjk,e^2=h,f^2=ik,i^2=j,j^2=k,b^a=bf,c^a=cg,c^b=cfk,d^c=dgj,e^b=ej,e^c=ek,f^a=fi,f^b=fij,f^c=fij,f^e=fk,g^a=gk,g^b=gk,h^b=hk,i^a=ij,i^b=ijk,i^c=ijk,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	#72227077, #72227158, #72227168
maximal normal subgroup of	#72227077, #72227158, #72227168

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