#72312968

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#3027
commutator	C₂×#41	C₂⁴×C₄
Frattini, cosocle	C₂²×#41	C₂⁵
socle	C₂²	C₂×#32926
prime radical	C₂×#511930	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

15705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^11|a^2=f,c^2=h,d^2=f,e^2=f,h^2=k,c^a=ck,c^b=cj,d^a=dg,d^b=di,d^c=dhk,e^c=ek,g^b=gj,h^d=hk,i^a=ijk,i^c=ik,i^g=ik,j^d=jk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313012
maximal normal subgroup of	#72313012

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