#72231721

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

795485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 16 points:

(6,8);(9,10)(11,15,16,14)(12,13);(11,14,16,15);(1,2)(3,4);(2,4)(5,6,7,8)(9,11)(10,14)(12,15)(13,16)

Polycyclic presentation:

2^11|b^2=jk,c^2=jk,e^2=g,h^2=k,e^a=eg,e^b=ei,e^c=eh,e^d=ef,h^b=hj,h^e=hk,i^b=ik,i^c=ijk,i^h=ik,j^e=jk

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234176, #72234590, #72252600
maximal normal subgroup of	#72234176, #72234590, #72252600
maximal quotient of	#72260800, #72282974
automorphism group of	#1296, #1619, #1620, …

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