#72216226

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₆ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈²	C₄²
commutator	C₄	C₂×C₈×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂³
socle	C₂²	C₂×#1311
prime radical	C₂³	#1311
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=g,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bhj,d^b=dj,e^a=ej

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216231, #72216326, #72216385, …
maximal normal subgroup of	#72216231, #72216326, #72216385, …
maximal quotient of	#72225076

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