#72339049

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72262790
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72213224	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35527 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^11|a^2=g,c^2=h,d^2=jk,e^2=h,g^2=k,b^a=bf,c^a=cjk,c^b=ce,d^a=di,d^b=dh,d^c=dghik,e^a=ei,e^b=eh,e^c=eh,e^d=ehjk,f^a=fh,f^c=fij,f^d=fjk,f^e=fjk,g^b=gh,g^f=gk,h^a=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	#72339051
maximal normal subgroup of	#72339051

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