#72258449

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₄	C₂⁴
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

23217 conjugacy classes of subgroups; only the 2461 normal subgroups are shown.

normal subgroups only

This group has 23217 conjugacy classes of subgroups — too many to draw. Only its 2461 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 256

Polycyclic presentation:

2^10|a^2=j,b^2=f,c^2=fg,c^b=cj,d^b=dgi,d^c=dgh,e^b=ehi,e^c=egi

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72258460
maximal subgroup of	#72258460, #72262503
maximal normal subgroup of	#72258460, #72262503

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