#72307526

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18186
commutator, Frattini, cosocle	C₂×#97	C₂⁴
prime radical	C₂×#493	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=gh,b^2=f,c^2=gh,d^2=gh,e^2=h,f^2=i,i^2=j,b^a=bf,c^b=chi,d^a=dgh,d^b=de,d^c=dgh,e^a=eg,e^b=egh,e^c=ej,e^d=ehj,f^a=fi,f^c=fj,f^d=fg,f^e=fj,g^b=gj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307544, #72307684, #72307737
maximal normal subgroup of	#72307544, #72307684, #72307737

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