#72343416

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72343272
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	#2108	C₂⁴
prime radical	#72216687	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622660 ◁ G
LECS	G ▷ #2108 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5351 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:

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

Polycyclic presentation:

2^11|a^2=e,d^2=ijk,e^2=hj,g^2=i,h^2=jk,i^2=jk,j^2=k,b^a=bg,c^a=cf,d^a=djk,d^b=dhk,d^c=dhij,e^b=eh,e^c=ehij,e^d=ek,f^a=fhij,f^d=fk,f^e=fk,g^a=ghik,g^b=gij,g^d=gk,h^b=hj,i^a=ik,i^b=ij,j^b=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343423, #72343482
maximal normal subgroup of	#72343423, #72343482

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