#72313401

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	#7623880	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ #20521 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2599 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=f,d^2=h,e^2=fghi,h^2=i,i^2=j,b^a=bgj,c^b=cg,d^a=dg,d^b=dj,d^c=dh,e^c=eh,e^d=egij,f^d=fj,f^e=fj,h^c=hi,h^e=hj,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313561, #72313732, #72314594
maximal normal subgroup of	#72313561, #72313732, #72314594

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