#72286751

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×#102
commutator	C₂²×C₈	C₂³×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂×#32930	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄² ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

15102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 512

Polycyclic presentation:

2^11|c^2=f,d^2=e,e^2=i,g^2=jk,j^2=k,c^a=cf,c^b=ch,d^c=dfgj,e^c=ek,g^c=gj,g^d=gjk,j^c=jk,j^d=jk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286863
maximal normal subgroup of	#72286863
maximal quotient of	#72287624

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);