#72314151

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#62251	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fhij,e^2=gj,f^2=j,h^2=i,i^2=j,d^b=dg,d^c=dh,e^a=egj,e^c=egj,e^d=egj,f^d=fj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314160, #72314408, #72315974
maximal normal subgroup of	#72314160, #72314408, #72315974

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