#72318473

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#35348	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5227 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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=hi,d^2=fij,g^2=j,b^a=bf,c^a=cfh,c^b=cg,d^a=dfij,e^d=ej,f^c=fj,g^a=gi,g^c=gij,h^b=hi,h^c=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318548, #72318756, #72319015
maximal normal subgroup of	#72318548, #72318756, #72319015

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