#72338728

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	#72237742
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #21749 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

2462 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)(3,11,15,12)(4,16)(5,9)(6,10)(7,14)(8,19,20,26)(13,24,32,31)(17,21)(18,27,29,30)(22,28)(23,25);(3,12)(4,17)(5,22)(7,23)(8,26)(9,28)(11,15)(14,25)(16,21)(18,29)(19,20)(24,31);(1,3,9,11)(2,7,5,14)(4,18,16,32)(6,15,28,12)(8,27,26,24)(10,25,22,23)(13,21,29,17)(19,31,20,30);(1,4,6,21)(2,8,10,20)(3,13,15,32)(5,19,22,26)(7,24,25,31)(9,17,28,16)(11,18,12,29)(14,30,23,27)

Polycyclic presentation:

2^11|a^2=gj,c^2=efjk,d^2=k,e^2=hik,f^2=hjk,b^a=bgj,c^a=ce,c^b=cefhjk,d^a=degh,d^b=defhik,d^c=defik,e^a=eh,e^b=eg,e^c=eghi,e^d=ehjk,f^a=fghj,f^b=fgk,f^c=fghijk,f^d=fhjk,f^e=fijk,g^c=gi,g^d=gik,g^e=gjk,g^f=gjk,h^a=hjk,h^b=hjk,h^c=hjk,h^d=hk,h^e=hk,i^a=ij,i^e=ik,i^f=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338805
maximal normal subgroup of	#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.7*F.10);;
gap> SetPower(coll, 3, F.5*F.6*F.10*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.8*F.9*F.11);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.8*F.9*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, 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> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);