#72283793

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20585 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^11|a^2=j,b^2=gj,c^2=eghik,e^2=hik,f^2=h,c^a=cf,c^b=ce,d^a=dhjk,d^b=dhijk,d^c=dhijk,e^a=eg,e^b=eh,e^c=eik,e^d=ek,f^a=fhk,f^b=fgik,f^c=fghk,f^d=fk,f^e=fik,g^c=gi,g^d=gk,g^e=gj,g^f=gjk,h^b=hj,h^c=hjk,h^e=hk,i^a=ij,i^b=ijk,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284259, #72338791, #72338805
maximal normal subgroup of	#72284259, #72338791, #72338805

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