#72210547

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#8541
commutator	C₂⁶	C₂²×C₄
Frattini, cosocle	C₂²×#120	C₂³
prime radical	#6342371	C₂
LCS	G ▷ C₂⁶ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

107051 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 107051 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|a^2=d,b^2=g,e^2=i,b^a=bdei,c^a=cf,c^b=ch,d^b=dg,d^c=dh,e^a=eg,e^b=egi,e^c=eh,e^d=ei,f^a=fh,f^b=fj,f^d=fj,f^e=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210568, #72215688, #72215789, …
maximal normal subgroup of	#72210568, #72215688, #72216682, …
maximal quotient of	C₂×#72210547, #72238276, #72241312, …
automorphism group of	#562

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