#72314038

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	#7623625	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2992 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)(11,17)(12,23,28,35)(13,30)(14,16)(15,20)(18,29,24,26)(19,27)(21,37,25,40)(22,32)(31,36)(33,38,34,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20)(12,24)(13,19)(14,16)(15,17)(18,28)(21,33)(22,36)(23,26)(25,34)(27,30)(29,35)(31,32)(37,38)(39,40);(2,6)(5,8)(10,16)(12,25,28,21)(13,31)(17,20)(18,34,24,33)(22,27)(23,37,35,40)(26,39,29,38);(3,7)(5,8)(9,11)(10,17)(12,26)(13,31)(14,15)(16,20)(18,23)(21,38)(24,35)(25,39)(28,29)(30,36)(33,40)(34,37);(2,6)(5,8)(9,12,27,38,11,21,13,26,14,28,22,39,15,25,31,29)(10,18,30,40,20,34,19,35,16,24,36,37,17,33,32,23)

Polycyclic presentation:

2^10|a^2=f,c^2=fhj,e^2=fgi,g^2=i,i^2=j,c^a=ch,c^b=cj,d^b=dh,e^a=eg,e^b=ej,e^c=ei,e^d=eghi,f^e=fj,g^a=gi,g^c=gj,g^d=gij,g^e=gj,i^a=ij,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314059, #72314408, #72314497, …
maximal normal subgroup of	#72314059, #72314408, #72314497, …

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