#72287529

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#489289	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #17320 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=gi,d^2=e,e^2=i,f^2=h,g^2=ij,h^2=j,c^a=cg,c^b=cf,d^a=dij,d^b=dij,d^c=dhj,e^c=ej,f^a=fij,f^b=fh,f^c=fhi,f^d=fj,g^a=gij,g^b=gij,h^b=hj,h^c=hj

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287545, #72320132, #72321057
maximal normal subgroup of	#72287545, #72320132, #72321057

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