#72258240

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂⁴	C₂²×C₂²⋊C₄
prime radical	C₂×#34738	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

17125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^10|b^2=efj,d^2=j,e^2=j,f^2=j,b^a=be,c^a=ch,c^b=cg,d^a=defij,d^b=dej,e^a=ej,e^d=ej,i^a=ij,i^d=ij

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258244, #72258257, #72259481, …
maximal normal subgroup of	#72258244, #72258257, #72259481, …
maximal quotient of	#72267304

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