#72226865

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1629
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂⁴	#453
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1459 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=f,e^2=h,g^2=ij,i^2=j,b^a=bfgi,c^a=cf,c^b=ceh,d^b=dj,e^a=ei,e^c=eh,g^a=gi,g^c=gij,h^a=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226881
maximal normal subgroup of	#72226881

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