#72227120

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄×#106
commutator	C₄×C₈	C₂³×C₈
Frattini, cosocle	C₄²×C₈	C₂⁴
prime radical	C₂²×D₈	#106
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈×D₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|a^2=e,b^2=eik,c^2=egj,d^2=gik,e^2=h,f^2=i,g^2=j,i^2=k,b^a=bf,c^a=cg,c^b=cfgik,d^a=dijk,d^b=dijk,d^c=dgik,e^b=ek,e^c=ej,f^a=fi,f^b=fi,f^c=fi,g^c=gj,i^a=ik,i^b=ik,i^c=ik

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227230
maximal normal subgroup of	#72227230
maximal quotient of	#72227121

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