#72235161

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	#124
commutator	C₂²×C₄	C₄³
Frattini, cosocle	C₂⁵×C₄	C₂³
socle	C₂⁴	#393
prime radical	C₂⁶	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵×C₄ ▷ C₂ ▷ C₁

At least 6068 conjugacy classes of subgroups; only the 3269 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=d,b^2=dgj,c^2=e,g^2=j,h^2=j,i^2=j,b^a=bh,c^a=cgij,c^b=cf,d^b=dj,e^b=ej,f^a=fj,f^c=fj,g^b=gj,h^b=hj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235189, #72235548
maximal normal subgroup of	#72235189, #72235548
maximal quotient of	#72235227, #72235683

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