#72313921

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#342
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3923 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=fgj,f^2=ij,g^2=j,b^a=bhi,c^a=chij,c^b=cefij,d^c=dj,e^a=ej,e^c=eh,f^b=fhi,f^c=fhj,f^e=fj,g^b=ghj,g^c=ghj,g^e=gj,h^b=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313929, #72313936, #72313950
maximal normal subgroup of	#72313929, #72313936, #72313950

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