#72287412

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285955
commutator	C₂²×C₁₆	C₂×C₄²
Frattini, cosocle	#33326	C₂³
prime radical	#3444	D₈
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622660 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^11|b^2=e,c^2=d,d^2=h,f^2=ij,i^2=jk,j^2=k,b^a=bg,c^a=chk,c^b=cefi,d^b=djk,e^a=ehk,f^a=fhk,f^b=fi,f^c=fij,f^d=fk,g^b=ghk,g^c=gk,g^e=gk,g^f=gk,h^b=hk,i^b=ij,i^c=ijk,j^b=jk,j^c=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287481, #72343118, #72343482
maximal normal subgroup of	#72287481, #72343118, #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, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.9*F.10);;
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.8*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*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);