#72338913

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72262779
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72236974	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35272 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

3822 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:

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

Polycyclic presentation:

2^11|a^2=h,c^2=hjk,d^2=ij,e^2=hj,g^2=k,b^a=bh,c^a=cg,c^b=ce,d^a=dghk,d^b=df,d^c=dgh,e^a=eh,e^b=ehj,e^c=ehj,e^d=ehi,f^a=fh,f^c=fij,f^d=fj,f^e=fjk,g^b=gh,g^c=gk,g^e=gk,g^f=gk,h^c=hk,h^d=hk,i^b=ij,i^c=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338932
maximal normal subgroup of	#72338932

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