#72223655

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#322
commutator	C₈	C₄²×C₈
Frattini, cosocle	C₂×#346	C₂³
socle	C₂²	C₂×#1313
prime radical	C₂³	C₂×#347
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=f,b^2=d,c^2=e,d^2=g,e^2=gj,f^2=h,h^2=i,i^2=j,b^a=bhj,c^a=chij,d^a=dij,e^a=ei,f^b=fij,f^c=fi,f^d=fj,f^e=fj,g^a=gj,h^b=hj,h^c=hj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223687, C₂×#72223655
maximal normal subgroup of	#72223687, C₂×#72223655
maximal quotient of	C₈×#7333, C₂×#72223655

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