#72314321

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#10577535	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5332 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)(11,16)(12,17)(13,15)(14,18)(19,27)(20,30)(21,31)(22,32)(23,33)(24,34)(25,28)(26,29)(35,38)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(11,19)(16,27)(20,24)(22,35)(30,34)(32,38)(36,37)(39,40);(1,3,5,4)(2,6,7,8)(10,15)(11,19)(17,28)(18,29)(20,24)(22,35)(31,33)(36,37);(2,7)(6,8)(11,20,19,24)(12,23)(14,26)(16,30,27,34)(17,33)(18,29)(21,25)(22,37,35,36)(28,31)(32,40,38,39);(1,4,5,3)(2,8,7,6)(9,11,21,36,26,35,25,20,13,19,23,37,14,22,12,24)(10,16,31,39,29,38,28,30,15,27,33,40,18,32,17,34)

Polycyclic presentation:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314329, #72314559, #72315551
maximal normal subgroup of	#72314329, #72314559, #72315551

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