#72258190

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₄×#442
commutator	C₄×C₈	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂⁴
socle	C₂³	#183994
prime radical	C₂²×D₁₆	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#346 ◁ #72216161 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

2797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 2048

A minimal faithful permutation representation — 4 generators on 40 points:

(26,33)(27,29)(28,39)(30,31)(34,37)(35,38)(36,40);(25,26)(27,34)(28,40)(29,35)(30,37)(31,36)(32,38)(33,39);(1,2,4,6,5,7,8,3)(9,10,11,15,12,17,20,24,13,18,19,16,21,14,22,23)(25,26,28,35,30,36,29,40,32,38,27,33,31,37,39,34);(1,3,8,7,5,6,4,2)(10,14,18,17)(11,19)(15,24,16,23)(20,22)(25,27,31,29,32,28,30,39)(26,34,36,35,38,40,37,33)

Polycyclic presentation:

2^12|c^2=e,d^2=efik,e^2=h,f^2=i,g^2=jl,h^2=k,i^2=k,j^2=l,b^a=bg,c^a=cg,c^b=cgj,d^a=dj,d^c=di,e^a=ej,e^b=ejl,e^d=ek,f^a=fjl,f^b=fjl,f^c=fk,f^d=fk,g^a=gj,g^b=gj,g^c=gl,g^d=gl,h^a=hl,h^b=hl,i^a=il,i^b=il,j^a=jl,j^b=jl

Conjugacy classes & character table

292 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72258194

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