#72260730

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15259
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6346728	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

3947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=g,c^2=h,d^2=g,f^2=j,b^a=be,c^a=cej,c^b=cgh,d^a=df,d^b=dg,e^b=ei,e^c=ej,e^d=eij,f^a=fj,f^b=fj,f^c=fij,f^d=fij,g^a=gi,h^a=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260753
maximal normal subgroup of	#72260753

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