#72338710

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	#72237728
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72216457	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ #21746 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

2759 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)(3,8,13,19)(4,14,17,25)(5,10)(6,7)(9,16)(11,15)(12,23,29,30)(18,21)(20,31)(22,24,27,26)(28,32);(5,18)(10,21)(12,29)(23,30);(1,3)(2,8,6,19)(4,15,17,32)(5,12,18,29)(7,13)(9,24)(10,23)(11,25,28,14)(16,27)(20,26)(21,30)(22,31);(1,4)(2,9)(3,11)(5,16)(6,20)(7,17)(8,22)(10,25)(12,26)(13,28)(14,21)(15,30)(18,31)(19,27)(23,32)(24,29)

Polycyclic presentation:

2^11|a^2=gk,c^2=ij,e^2=hi,f^2=hjk,c^a=cghik,d^a=de,d^b=dhk,d^c=defhjk,e^a=eh,e^b=ej,e^c=eg,e^d=ehi,f^a=fghj,f^c=fgk,f^d=fhi,f^e=fijk,g^c=gk,g^d=gi,g^e=gjk,g^f=gjk,h^a=hjk,h^c=hjk,h^e=hk,i^a=ij,i^e=ik,i^f=ik,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.11);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.8*F.10*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.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
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, 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, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);