#72314182

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 ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3326 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,23)(13,16)(14,18)(15,20)(17,25)(21,29)(22,33)(24,32)(26,31)(27,30)(28,34)(35,36)(37,40)(38,39);(1,3,5,2)(4,6,8,7)(9,11,13,20)(10,15,16,19)(14,27,26,22)(17,23)(18,33,31,30)(24,28)(29,35)(37,38);(2,3)(6,7)(10,16)(11,20)(12,24,25,28)(14,22)(17,32,23,34)(18,30)(21,37,36,38)(26,27)(29,39,35,40)(31,33);(2,3)(4,8)(9,12,22,37,20,36,14,28,13,25,27,38,11,21,26,24)(10,17,33,39,15,29,18,32,16,23,30,40,19,35,31,34);(1,4,5,8)(2,6,3,7)

Polycyclic presentation:

2^10|b^2=f,c^2=fg,d^2=fghi,e^2=gj,h^2=i,i^2=j,b^a=bg,c^a=cg,c^b=cg,d^a=dg,d^b=dgi,d^c=dh,e^a=egj,e^b=egj,e^d=egj,f^d=fj,h^b=hj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314202, #72314251, #72314329, …
maximal normal subgroup of	#72314202, #72314251, #72314329, …

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