#72237314

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#1296
commutator	C₂×C₄²	C₈²
Frattini, cosocle	C₂×C₄⁴	C₂²
socle	C₂³	#7025
prime radical	C₂⁵	#342
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂×C₄⁴ ◁ G
LECS	G ▷ C₂×C₄⁴ ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=c,b^2=d,c^2=f,d^2=g,e^2=j,h^2=k,b^a=be,c^b=ci,d^a=dh,e^a=eij,e^b=ehj,g^a=gk,h^a=hk,j^b=jk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72237315

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