#72281719

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂⁵×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁷
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#3449 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Only the 34373 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|b^2=j,c^2=j,h^2=j,b^a=bj,d^c=dh,e^a=ej,e^b=ei,e^d=ej,f^b=fj,f^c=fj,g^d=gj,h^c=hj,h^d=hj

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72283092

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