#72313833

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	D₄×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20493 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3103 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=f,d^2=f,e^2=h,h^2=i,i^2=j,b^a=bg,c^b=cj,d^b=dj,d^c=dgj,e^a=ei,e^c=eh,e^d=ei,f^e=fj,h^a=hj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313898, #72314059, #72314202, …
maximal normal subgroup of	#72313898, #72314059, #72314202, …

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