#72314077

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	#7623719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314559, #72315212
maximal normal subgroup of	#72314098, #72314559, #72315212

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