#72313271

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	#7623360	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3004 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)(9,10,13,16)(11,19,20,15)(12,17,25,32)(14,18,27,29)(21,36,37,30)(22,28,38,31)(23,39,40,33)(24,35,26,34);(1,2)(3,6)(4,8)(5,7)(9,11,13,20)(10,15,16,19)(12,23)(14,28,27,31)(17,33)(18,22,29,38)(21,35)(24,36)(25,40)(26,30)(32,39)(34,37);(3,4)(6,8)(10,16)(11,20)(12,24)(14,29)(17,34)(18,27)(21,33)(22,28)(23,30)(25,26)(31,38)(32,35)(36,40)(37,39);(1,3)(2,6)(4,5)(7,8)(9,10,13,16)(11,19,20,15)(14,29,27,18)(22,31,38,28)(23,40)(24,26)(33,39)(34,35);(1,4,5,3)(2,6,7,8)(9,12,14,24,10,17,18,35,13,25,27,26,16,32,29,34)(11,21,22,39,19,36,28,40,20,37,38,33,15,30,31,23)

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=f,e^2=fgh,h^2=i,i^2=j,b^a=bg,c^a=cj,c^b=cj,d^a=dgj,d^b=dj,d^c=dg,e^b=eghij,e^c=egh,e^d=egi,f^e=fj,h^b=hi,h^c=hij,h^d=hj,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313345, #72314160, #72314604
maximal normal subgroup of	#72313345, #72314160, #72314604

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