#72256629

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂⁴×C₄
Frattini, cosocle	C₂²×D₄	C₂⁵
socle	C₂²	C₂²×B(4,𝔽₂)
prime radical	C₂×#60387	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

16115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^10|a^2=j,c^2=fh,d^2=ij,g^2=j,c^b=cf,d^b=dg,e^d=ej,f^d=fi,g^b=gj,g^c=gij,g^f=gj,h^d=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256634, #72256658, #72256666
maximal normal subgroup of	#72256634, #72256658, #72256666
maximal quotient of	#72263741, #72263746

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