#72287739

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623520	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=j,d^2=f,e^2=fh,f^2=h,g^2=i,i^2=j,b^a=bh,c^b=chj,d^a=dh,d^b=di,d^c=dh,e^a=eh,e^b=eghi,e^c=ej,f^b=fj,g^d=gj,g^e=gi,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287790, #72319199, #72319265, …
maximal normal subgroup of	#72287790, #72319199, #72319265, …

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