#72257686

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#95
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₄³	C₂⁴
prime radical	C₂⁵	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

2875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,e^2=h,f^2=i,g^2=j,b^a=be,c^a=ch,c^b=ci,d^b=di,e^a=ei,e^b=ej,f^a=fhj,g^b=ghi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257692, #72257764, #72257869
maximal normal subgroup of	#72257692, #72257764, #72257869

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