#72238830

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×#120
commutator	C₂×C₄²	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁵
socle	C₂³	C₂²×#379
prime radical	C₂⁶	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

33620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=h,c^2=fhik,e^2=jk,f^2=jk,g^2=j,i^2=k,b^a=bg,c^a=ci,c^b=cghjk,d^a=djk,e^a=ek,e^b=ejk,e^c=ejk,f^b=fj,f^c=fj,g^b=gj,g^c=gj,i^a=ijk,i^b=ij,i^c=ij

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238833, #72238851
maximal normal subgroup of	#72238833, #72238851

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