#72313756

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	#7623360	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20473 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2842 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fgj,d^2=hi,e^2=fj,h^2=i,i^2=j,b^a=bj,c^a=cj,c^b=cj,d^a=dgj,d^b=dghij,d^c=dghi,e^c=eg,e^d=egij,f^d=fj,h^b=hi,h^c=hi,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313816, #72314497, #72314604
maximal normal subgroup of	#72313816, #72314497, #72314604

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