#72287331

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₁₆
Frattini, cosocle	C₂×#1311	C₂⁴
prime radical	C₂×#13661	C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#7053 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^12|b^2=e,c^2=eh,d^2=f,e^2=h,f^2=i,h^2=j,i^2=k,k^2=l,d^a=dg,d^b=df,d^c=di,e^d=ekl,f^b=fi,f^c=fk,f^e=fl,h^d=hl,i^b=ik,i^c=il,k^b=kl

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72287333

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);