#72210602

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄×C₈² · Schur multiplier (H₂) = C₂×C₈²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₄²
commutator	C₄	C₄×C₈²
Frattini, cosocle	C₄²×C₈	C₂³
socle, prime radical	C₂³	#1726
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=j,c^a=ch,d^c=dj,f^a=fj

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72234317, #72234359
maximal subgroup of	#72210664, #72210694, #72234254
maximal normal subgroup of	#72210664, #72210694, #72234254
maximal quotient of	#72234231, #72234273

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