#72225227

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#9837
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle	C₂²	#102117
prime radical	#491485	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6,7,11,8,15,9,16)(3,10)(12,14)(18,22,25,26)(19,27)(23,29,31,30)(24,32);(1,2)(3,11)(4,7)(5,8)(6,14)(9,13)(10,16)(12,15)(18,23)(19,24)(20,28)(22,29)(25,31)(26,30)(27,32);(17,18,24,29,20,26,27,31,21,25,32,30,28,22,19,23)

Polycyclic presentation:

2^11|a^2=d,c^2=fi,d^2=g,e^2=h,f^2=ik,h^2=j,i^2=k,b^a=be,c^a=cik,c^b=cf,d^b=dh,d^c=dk,e^a=ej,e^b=ehj,f^a=fk,f^b=fi,g^b=gj,h^b=hj,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225236, #72225255
maximal normal subgroup of	#72225236, #72225255
maximal quotient of	#72225255

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