#72237306

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄
commutator	C₂×C₄	C₄²×C₈
Frattini, cosocle	C₂³×C₄²	C₂³
socle	C₂⁴	C₂×#95
prime radical	C₂⁵	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

8121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=h,d^2=i,g^2=j,b^a=bg,c^a=cf,e^a=ej,g^a=gj

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237314, #72240065, #72240069
maximal normal subgroup of	#72237314, #72240065, #72240069
maximal quotient of	#72237308

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