#72235540

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	#121
commutator	C₂²×C₄	C₄³
Frattini, cosocle	C₂⁵×C₄	C₂³
socle	C₂⁴	#388
prime radical	C₂⁶	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵×C₄ ▷ C₂ ▷ C₁

At least 6034 conjugacy classes of subgroups; only the 3269 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=defhj,d^2=j,g^2=j,h^2=j,i^2=j,b^a=bf,c^a=cfij,c^b=cfgj,d^b=dj,f^a=fj,f^c=fj,g^a=gj,g^b=gj,g^c=gj,h^a=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235548, #72238822, #72238847
maximal normal subgroup of	#72235548, #72238822, #72238847
maximal quotient of	#72235632, #72235683, #72238825

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