#72320400

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	#489287	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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=egi,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bg,c^a=cf,c^b=cfi,d^b=di,d^c=dhj,e^c=ej,f^a=fh,f^b=fh,f^c=fhj,f^d=fj,g^a=gi,h^a=hj,h^b=hj,h^c=hj

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320455, #72320975
maximal normal subgroup of	#72320455, #72320975

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