#72227240

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#231631
commutator	C₈²	C₂³×C₄
Frattini, cosocle	C₂×C₈²	C₂⁴
prime radical	#72213956	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=fgij,d^2=gh,e^2=ijk,f^2=hijk,g^2=ij,h^2=j,i^2=k,b^a=bf,c^a=cfgijk,c^b=cfghik,d^a=dhij,d^b=dghik,e^b=ei,e^c=ej,e^d=ek,f^a=fh,f^b=fhi,g^a=gk,g^b=gijk,h^a=hj,h^b=hj,i^b=ik

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227243

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