#72265615

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gi,c^2=g,e^2=hi,f^2=i,h^2=j,i^2=j,b^a=bf,c^b=ch,d^b=dhi,e^a=eghi,e^d=ej,f^a=fi,f^b=fi,f^c=fj,f^e=fj,g^b=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265645, #72307751
maximal normal subgroup of	#72265645, #72307751

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