#72321346

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₄²	C₂⁶
socle	C₂²	C₂³×C₄×D₄
prime radical	#10576902	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #60280 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

11681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^10|a^2=g,b^2=ghj,e^2=i,f^2=hj,g^2=i,h^2=j,b^a=bh,d^a=dj,d^b=dij,e^a=ej,e^b=eij,e^c=ej,e^d=eij,f^a=fhj,f^b=fj,h^a=hj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321406, #72322376
maximal normal subgroup of	#72321406, #72322376

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