#72265422

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#8350
commutator	C₂×#97	C₂²×C₄
Frattini, cosocle	C₂×#405	C₂³
prime radical	#7613557	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^10|c^2=efj,d^2=hj,e^2=ij,f^2=j,g^2=j,h^2=j,i^2=j,c^a=ce,c^b=cd,d^a=dg,d^b=dh,d^c=dj,e^a=ei,e^b=egj,e^d=ej,f^a=fj,f^b=fgh,f^d=fj,g^a=gj,g^b=gj,g^c=gj,h^a=hj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265427, #72307241, #72307255
maximal normal subgroup of	#72265427, #72307241, #72307255

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