#72314537

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	#7623257	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2548 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=hj,d^2=fj,e^2=ghj,g^2=i,i^2=j,c^a=ch,d^b=dhj,e^a=ehij,e^b=ej,e^c=ej,e^d=eg,f^e=fj,g^a=gj,g^d=gi,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314559, #72314604, #72314619
maximal normal subgroup of	#72314559, #72314604, #72314619

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