#72313541

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁴×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁶
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

43747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^10|c^2=j,e^2=g,f^2=hij,g^2=j,e^a=ej,e^d=eg,f^a=fj,f^b=fh,f^c=fj,f^d=fghi,f^e=fgj,g^d=gj,g^f=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315559
maximal normal subgroup of	#72315559

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