#72292786

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₈
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁸	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

145101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^10|d^2=e,e^2=i,d^a=dh,d^b=df,d^c=dfg,e^c=ej,g^d=gj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72292799, #72292816

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