#72224858

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂⁴
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,e^2=h,f^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ce,d^a=dj,e^a=ej,e^c=eh,f^b=fj,f^c=fi,g^a=gj,g^b=gj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224866, #72224934, #72226811
maximal normal subgroup of	#72224866, #72224934, #72226811

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