#72314072

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	#7623714	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2762 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)(11,16)(12,17)(13,18)(14,15)(19,32)(20,30)(21,31)(22,26)(23,27)(24,29)(25,28)(33,37)(34,40)(35,39)(36,38);(3,4)(6,8)(10,15)(11,19,22,24)(12,21)(13,25)(16,29,26,32)(17,33)(20,34,36,35)(23,37)(27,31)(30,39,38,40);(1,3)(2,6)(4,5)(7,8)(10,15)(16,26)(17,27)(18,28)(29,32)(30,38)(31,33)(39,40);(1,4,5,3)(2,6,7,8)(11,20,22,36)(12,23)(16,30,26,38)(17,27)(19,34,24,35)(21,37)(29,39,32,40)(31,33);(9,11,21,34,25,36,12,24,14,22,37,35,13,20,23,19)(10,16,31,40,28,38,17,29,15,26,33,39,18,30,27,32)

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=fg,e^2=fhi,h^2=i,i^2=j,b^a=bj,c^a=cgj,c^b=cg,d^b=dg,d^c=dg,e^b=eh,e^d=ei,f^e=fj,h^b=hi,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314329, #72315187
maximal normal subgroup of	#72314098, #72314329, #72315187

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