#72313860

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623358	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3084 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fj,d^2=fj,e^2=ghi,h^2=i,i^2=j,b^a=bgj,c^a=cj,c^b=cj,d^a=dgj,d^b=dgj,d^c=dg,e^b=eij,e^c=eghij,e^d=egij,f^e=fj,h^b=hj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313898, #72314160, #72315244
maximal normal subgroup of	#72313898, #72314160, #72315244

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