#72257673

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#386
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₄³	C₂⁴
prime radical	C₂×#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

2119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=g,e^2=hj,f^2=i,g^2=j,b^a=bf,c^a=chij,c^b=ce,d^b=di,e^a=eh,e^b=ej,f^a=fi,f^c=fh,g^b=ghj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257692
maximal normal subgroup of	#72257692

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