#72314180

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	#7623471	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20476 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2640 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=h,e^2=g,g^2=i,i^2=j,b^a=bh,d^a=dh,d^b=dj,d^c=dj,e^a=ej,e^b=ehij,e^c=eg,e^d=egi,f^e=fj,g^b=gj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314251, #72314408, #72314594
maximal normal subgroup of	#72314251, #72314408, #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, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);