#72262434

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15257
commutator, Frattini, cosocle	C₂⁶	C₂⁴
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

9961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Polycyclic presentation:

2^10|b^2=ef,c^2=fhj,d^2=j,b^a=be,c^a=ceg,c^b=cegh,d^a=dh,d^b=dfhij,d^c=dfhij,e^c=ei,e^d=ej,f^c=fj,g^b=gi,g^c=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262436
maximal normal subgroup of	#72262436

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