#72225238

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#8443
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
prime radical	C₂×D₈²	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

3010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|c^2=d,d^2=g,e^2=hj,f^2=ik,h^2=j,i^2=k,c^a=cf,c^b=ce,d^a=dk,d^b=dj,e^a=ei,e^b=eh,e^c=ehj,f^a=fi,f^b=fi,f^c=fik,h^a=hk,h^b=hj,h^c=hj,i^a=ik,i^b=ik,i^c=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225289
maximal normal subgroup of	#72225289
maximal quotient of	#72225239

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