#72338720

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237594
commutator	#379	C₂³×C₄
Frattini, cosocle	#1781	C₂⁴
prime radical	#72338716	C₂
LCS	G ▷ #379 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ #6374210 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

4587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=ghik,c^2=efhjk,e^2=hij,f^2=hjk,c^a=ce,c^b=chjk,d^a=dhik,d^c=dik,e^a=eh,e^b=ej,e^c=eg,e^d=ek,f^a=fgij,f^c=fg,f^e=fik,g^a=gk,g^c=gi,g^d=gk,g^e=gjk,g^f=gj,h^a=hjk,h^c=hjk,h^e=hk,i^a=ij,i^c=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338805, #72338824
maximal normal subgroup of	#72338805, #72338824

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