#72307805

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#105987
commutator	C₂³×C₈	C₂×C₄²
Frattini, cosocle	C₂³×#131	C₂³
prime radical	C₂³×#41	C₂×D₄
LCS	G ▷ C₂³×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂³×#131 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(9,10,15,19)(11,14,22,16)(13,28,30,18)(17,25)(20,36)(21,29,33,31)(24,37)(27,35)(32,38)(34,40);(1,2)(3,5,7,8)(4,6)(9,11,15,22)(10,14,19,16)(12,23,26,39)(13,21,30,33)(17,32,35,40)(18,31,28,29)(20,37,36,24)(25,38,27,34);(1,3)(2,5,6,8)(4,7)(9,12,18,34,11,20,29,40,15,26,28,38,22,36,31,32)(10,17,33,39,16,27,13,24,19,35,21,23,14,25,30,37)

Polycyclic presentation:

2^11|a^2=i,b^2=f,c^2=dejk,e^2=j,g^2=j,j^2=k,c^a=cg,c^b=cd,d^a=dk,d^b=dh,d^c=dhi,e^a=ej,e^b=ei,e^c=ehi,e^d=ek,f^c=fh,g^a=gj,g^b=gk,i^c=ik,j^a=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308179
maximal normal subgroup of	#72308179

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