#72315182

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#62328	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|e^2=f,f^2=ij,g^2=ij,i^2=j,b^a=bh,c^a=cj,c^b=chj,d^c=dhj,e^d=eg,f^d=fi,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315187, #72315286, #72315974
maximal normal subgroup of	#72315187, #72315286, #72315974

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