#72307937

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23848
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#576	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=fhj,d^2=hij,e^2=i,f^2=ij,h^2=j,i^2=j,c^a=ce,c^b=cefi,d^a=dghij,d^b=dghij,d^c=dhi,e^a=ei,e^b=ej,e^d=ej,f^a=fij,f^b=fj,g^c=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307984
maximal normal subgroup of	#72307984

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