#72340603

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1790
commutator	C₂×C₄⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
prime radical	#7710883	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=j,d^2=hij,e^2=h,g^2=j,c^a=ce,c^b=cj,d^a=dg,d^c=dh,e^a=ei,e^c=eh,e^d=eij,f^c=fhi,f^e=fj,g^a=gj,g^c=gi,g^e=gj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340605, #72340615, #72340734
maximal normal subgroup of	#72340605, #72340615, #72340734

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