#72224937

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#418918
commutator, Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
LCS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #181888 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

7823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=g,c^2=e,d^2=f,e^2=i,g^2=j,j^2=k,b^a=bh,c^a=cjk,c^b=cf,d^a=dgjk,d^b=df,d^c=def,e^a=ek,e^d=ei,f^a=fk,g^b=gk,g^c=gk,g^d=gj,h^a=hk,h^c=hk,j^d=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226832
maximal normal subgroup of	#72226832

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