#72310762

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#107114
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂³
prime radical	C₂×#3498	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

3051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=h,c^2=g,f^2=j,b^a=bf,c^a=ce,d^b=dh,d^c=di,e^a=ei,e^b=eh,e^c=eh,f^a=fhj,f^c=fhj,f^d=fj,f^e=fj,g^a=gh,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310767, #72311042, #72311073
maximal normal subgroup of	#72310767, #72311042, #72311073

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